Celebratio Mathematica — Askey — Complete Bibliography

[1] R. Askey and I. Hirschman, Jr.: “Weighted quadratic norms and ultraspherical polynomials, I,” Trans. Am. Math. Soc. 91 : 2 (May 1959), pp. 294–313. MR 107772 Zbl 0199.46701 article

[2] R. A. Askey: Mean convergence of orthogonal series and conjugate series. Ph.D. thesis, Princeton University, 1961. Advised by S. Bochner. MR 2613322 phdthesis

[3] R. Askey and I. I. Hirschman, Jr.: “Mean summability for ultraspherical polynomials,” Math. Scand. 12 (1963), pp. 167–177. MR 164198 Zbl 0132.29501 article

[4] R. Askey and S. Wainger: “On the behavior of special classes of ultraspherical expansions, II,” J. Analyse Math. 15 (1965), pp. 193–220. article

[5] R. Askey: “On some problems posed by Karlin and Szegö concerning orthogonal polynomials,” Boll. Un. Mat. Ital. (3) 20 : 1 (1965), pp. 125–127. MR 177148 Zbl 0127.03102 article

[6] R. Askey and S. Wainger: “Mean convergence of expansions in Laguerre and Hermite series,” Am. J. Math. 87 : 3 (July 1965), pp. 695–708. MR 182834 Zbl 0125.31301 article

[7] R. Askey: “Orthogonal expansions with positive coefficients,” Proc. Am. Math. Soc. 16 : 6 (1965), pp. 1191–1194. Part II was published in SIAM J. Math. Anal. 2:3 (1971). MR 185331 Zbl 0136.05103 article

[8] R. Askey and S. Wainger: “On the behavior of special classes of ultraspherical expansions, I,” J. Analyse Math. 15 (1965), pp. 193–220. MR 193290 Zbl 0132.29403 article

[9] R. Askey: “Orthogonal expansions of averaged functions,” J. Math. Anal. Appl. 14 : 2 (May 1966), pp. 326–331. MR 188699 Zbl 0136.37003 article

[10] R. Askey and S. Wainger: “Integrability theorems for Fourier series,” Duke Math. J. 33 : 1 (1966), pp. 223–228. MR 192260 Zbl 0136.36501 article

[11] R. Askey: “Norm inequalities for some orthogonal series,” Bull. Am. Math. Soc. 72 : 5 (1966), pp. 808–823. MR 198114 Zbl 0173.06703 article

[12] R. Askey and S. Wainger: “A transplantation theorem between ultraspherical series,” Ill. J. Math. 10 : 2 (1966), pp. 322–344. MR 211187 Zbl 0135.27603 article

[13] R. Askey and S. Wainger: “A transplantation theorem for ultraspherical coefficients,” Pac. J. Math. 16 : 3 (1966), pp. 393–405. MR 217508 Zbl 0136.37201 article

Let \( f(\theta) \) be integrable on \( (0,\pi) \) and define \begin{align*} a_n &= \int_0^{\pi} f(\theta)\cos n\theta \,d\theta,\\ b_n &= n^{1/2}\int_0^{\pi} f(\theta)P_n(\cos \theta)(\sin \theta)^{1/2} \,d\theta \end{align*} where \( P_n(x) \) is the Legendre polynomial of degree \( n \). Then \begin{equation*}\tag{1} c \leq \sum_{n=0}^{\infty}|a_n|^p (n+1)^{\alpha}/\sum_{n=0}^{\infty}|b_n|^p(n+1)^{\alpha} \leq C \end{equation*} for \( 1 < p < \infty \), \( -1 < \alpha < p - 1 \), where \( C \) and \( c \) depend on \( p \) and \( \alpha \) but not on \( f \). From this we obtain a form of the Marcinkiewicz multiplier theorem for Legendre coefficients. Also an analogue of the Hardy–Littlewood theorem on Fourier coefficients of monotone coefficients is obtained. In fact, any norm theorem for Fourier functions can be transplanted by (1) to a corresponding theorem for Legendre coefficients.

Actually, the main theorem of this paper deals with ultraspherical coefficients and (1) is just a typical special case, which is stated as above for simplicity.

[14] R. Askey: “Smoothness conditions for Fourier series with monotone coefficients,” Acta Sci. Math. (Szeged) 28 : 1–2 (1967), pp. 169–171. MR 212474 Zbl 0172.34701 article

[15] R. Askey: “A transplantation theorem for Jacobi coefficients,” Pac. J. Math. 21 : 3 (1967), pp. 393–404. MR 217509 Zbl 0172.08601 article

Let \( f(\theta) \) be integrable on \( (0,\pi) \) and define \[ a_n^{\alpha,\beta} = t_n^{\alpha,\beta}\int_0^{\pi} f(\theta)P_n^{(\alpha,\beta)}(\cos \theta) \Bigl(\sin \frac{\theta}{2}\Bigr)^{\alpha+(1/2)} \Bigl(\cos \frac{\theta}{2}\Bigr)^{\beta+(1/2)} d\theta \] where \( P_n^{(\alpha,\beta)}(x) \) is the Jacobi polynomial of degree \( n \), order \( (\alpha,\beta) \) and \[ [t_n^{\alpha,\beta}]^2 = \frac{(2n + \alpha + \beta + 1)\,\Gamma(n + 1)\,\Gamma(n + \alpha + \beta + 1)} {\Gamma(n + \alpha + 1)\,\Gamma(n + \beta + 1)} \] Then if \( \alpha \,\beta \, \gamma, \delta \geq -1/2 \) we have \[ \sum_{n=0}^{\infty} \bigl|a_n^{(\gamma,\delta)}\bigr|^p (n + 1)^{\sigma} \leq A\sum_{n=0}^{\infty}\bigl|a_n^{(\alpha,\beta)}\bigr|^p (n + 1)^{\sigma} \] for \( 1 < p < \infty \), \( -1 < \sigma < p - 1 \) whenever the right hand side is finite.

From this result any norm inequality for Fourier coefficients can be transplanted to give a corresponding norm inequality for Fourier–Jacobi coefficients.

[16] R. Askey and R. P. Boas, Jr.: “Fourier coefficients of positive functions,” Math. Z. 100 : 5 (1967), pp. 373–379. MR 219972 Zbl 0175.06303 article

[17] R. Askey: “Product of ultraspherical polynomials,” Am. Math. Monthly 74 : 10 (December 1967), pp. 1221–1222. MR 223618 Zbl 0153.09202 article

[18] R. Askey and S. Wainger: “A dual convolution structure for Jacobi polynomials,” pp. 25–36 in Orthogonal expansions and their continuous analogues (Edwardsville, IL, 27–29 April 1967). Edited by D. T. Haimo. Southern Illinois University Press (Carbondale, IL), 1968. MR 232027 Zbl 0174.36305 incollection

Deborah Tepper Haimo	Related
Stephen Wainger	Related

[19] R. Askey: “Jacobi polynomial expansions with positive coefficients and imbeddings of projective spaces,” Bull. Am. Math. Soc. 74 : 2 (1968), pp. 301–304. To I. J. Schoenberg on his 65th birthday. MR 220987 Zbl 0167.35003 article

[20] R. Askey: “\( L^p \) behavior of power series with positive coefficients,” Proc. Am. Math. Soc. 19 : 2 (1968), pp. 303–305. MR 223786 Zbl 0174.35304 article

[21] R. Askey and J. Fitch: “Positivity of the Cotes numbers for some ultraspherical abscissas,” SIAM J. Numer. Anal. 5 : 2 (June 1968), pp. 199–201. MR 228166 Zbl 0169.08301 article

Let \( P_n^{(\nu)}(x) \) be the ultraspherical polynomial of degree \( n \), order \( \nu \), \( \nu > -\frac{1}{2} \). Let \( x_k = x_{k,n} \) be the zeros of \( P_n^{(\nu)}(x) \) in decreasing order. Define \( \lambda_k = \lambda_{k,n} \) by \[ \sum_{k=1}^n \lambda_k x_k^r = \int_{-1}^1 x^r dx, \quad r = 0,1,\dots,n-1. \] Define \[ Q_n(f)=\sum_{k=1}^n \lambda_{k,n}f(x_{k,n}) .\] Szegő [1959, Theorem 15.5] has shown that \[ \lim_{n\to\infty}Q_n(f) = \int_{-1}^1 f(x)\,dx \] for an arbitrary Riemann integrable function for \( \nu \leq 2 \) by showing that \( \lambda_{k,n} \geq 0 \) for all sufficiently large \( n \). For \( -1/2 < \nu \leq 1/2 \) and \( 1 \leq \nu \leq 3/2 \) he has shown that \( \lambda_{k,n} > 0 \) for all \( n \), and he states that it is very probable that \( \lambda_k > 0 \) for all values of \( n \) provided that \( -1/2 < \nu \leq 2 \). We show that this is so for \( 1/2 < \nu \leq 2 \), which, combined with Szegő’s result, gives his conjecture.

Our proof uses a fraction integration formula connecting two ultraspherical polynomials. It follows from known results on hypergeometric functions, but a direct proof in this case is so easy that we give it.

[22] R. Askey, J. Fitch, and G. Gasper: “On a positive trigonometric sum,” Proc. Am. Math. Soc. 19 : 6 (December 1968), pp. 1507. MR 231107 Zbl 0174.35704 article

James S. Fitch	Related
George Gasper, Jr.	Related

[23] R. Askey: “Dual equations and classical orthogonal polynomials,” J. Math. Anal. Appl. 24 : 3 (December 1968), pp. 677–685. MR 234235 Zbl 0185.12601 article

In the last ten years there has been much interest in dual integral equations and a discrete analogue, dual series equations. Much of this work has been collected by Sneddon and it appears in [Sneddon 1966]. Any harmonic analyst looking at this work is struck by the absence of work on what might be called dual dual series equations, but which we will call dual sequence equations. A typical example is \begin{align*} a_n &= \int_0^{\pi}\sin\theta f(\theta)\cos n\theta \,d\theta,\quad n = 0,1,\dots,N,\\ a^{\prime}_n &= \int_0^{\pi}f(\theta)\cos n\theta \,d\theta,\quad n = N+1,\dots . \end{align*} We show how a number of such equations can be solved by dualizing methods developed by others for dual series equations.

[24] R. Askey: “Book review, V. Mangulis, ‘Handbook of series for scientists and engineers’,” J. Franklin Inst. 287 : 6 (July 1969), pp. 530–531. article

[25] R. Askey: Orthogonal polynomials and positivity. Technical report TW 112, Mathematisch Centrum, Amsterdam, August 1969. Later published in Special functions and wave propagation (1970). MR 247356 Zbl 0188.12402 techreport

The literature of special functions is full of explicit formulas, some extremely useful and some of no value at all. While much of the rest of mathematics is no longer as constructive as it was in the eighteenth century, this is not true in the theory of special functions. Even the extremely interesting recent work on Lie theory has been primarily concerned with specific formulas and has concentrated on general methods to obtain these formulas. I have no quarrel with this general tendency, since it is usually possible to prove more if a specific formula is obtained than if nothing more than an existence theorem is proven. However, there are times when the specific formulas obtained are not useful for certain problems, while a general property, say the positivity of some kernel, is the essential fact that is needed. We will give some instances of this, at times when we can not find a specific formula at all and other examples when a specific formula can be found but the positivity must be obtained by a different argument.

[26] R. Askey: “Mehler’s integral for \( P_n(\cos \theta) \),” Am. Math. Monthly 76 : 9 (1969), pp. 1046–1049. MR 1535644 Zbl 0182.08301 article

[27] R. Askey and J. Fitch: “Integral representations for Jacobi polynomials and some applications,” J. Math. Anal. Appl. 26 : 2 (May 1969), pp. 411–437. MR 237847 Zbl 0172.08803 article

An integral for \[ \frac{P_n^{(\alpha + \mu,\beta)}(x)}{P_n^{(\alpha + \mu,\beta)}(1)} \] in terms of \[ \frac{P_n^{(\alpha,\beta)}(y)}{P_n^{(\alpha,\beta)}(1)} \] with a positive kernel is obtained. For \( \beta = \pm 1/2 \) this integral is equivalent to an important integral of Feldheim and Vilenkin connecting ultraspherical polynomials. As an application we show that \[ \frac{P_n^{(\alpha,\alpha)}(x)}{P_n^{(\alpha,\alpha)}(1)} = \int_{-1}^1 \frac{P_n^{(\beta,\beta)}(y)}{P_n^{(\beta,\beta)}(1)}d\mu(y) \] where \( \alpha > \beta \geq 1/2 \), \( -1 \leq x \leq 1 \), and \( d\mu(y) \) is a positive measure which depends on \( x \) but not \( n \). For \( \beta = -1/2 \) this is a result of Seidel and Szasz. Similar results are obtained for Jacobi polynomials and the positivity of certain sums of ultraspherical and Jacobi polynomials is obtained.

[28] R. Askey: “A transplantation theorem for Jacobi series,” Ill. J. Math. 13 : 3 (1969), pp. 583–590. MR 246038 Zbl 0174.35303 article

[29]R. Askey: Mean convergence of orthogonal series and Lagrange interpolation. Technical report TW 113, Mathematisch Centrum, Amsterdam, September 1969. A corrected version of this was later published in Acta Math. Acad. Sci. Hung. 23:1–2 (1972). MR 259473 techreport

[30] R. Askey: Three notes on orthogonal polynomials. Technical report TW 114/69, Mathematisch Centrum, Amsterdam, October 1969. MR 261049 techreport

[31] R. Askey and S. Wainger: “A convolution structure for Jacobi series,” Am. J. Math. 91 : 2 (April 1969), pp. 463–485. MR 264132 Zbl 0186.12303 article

For ultraspherical polynomials and in particular for Legendre polynomials there is an addition formula which has as a consequence a type of convolution. This was introduced by Gelfand [1950] and by Bochner [1954] and has a number of very interesting consequences. It has been used to obtain the analogue of the Hardy–Littlewood theorem on fractional integration [Askey and Wainger 1965; Muckenhoupt and Stein 1965] and to obtain a simple proof of Kogbetliantz’s theorem on uniform \( (C,\alpha) \) summability from the end point result that Szegő has in his book [1959]. For Jacobi polynomials the addition theorem has not been found yet, and so the analogous convolution structure is not known. We find a form of this convolution which is sufficient to obtain the above results as well as other applications.

[32] R. Askey and J. Fitch: “A trigonometric inequality (J. N. Lyness and C. B. Moler),” SIAM Review 11 : 1 (1969), pp. 82–86. article

[33] R. Askey: Eight lectures on orthogonal polynomials. Technical report TC 51/70, Mathematisch Centrum, Amsterdam, April 1970. techreport

[34] R. Askey: “Orthogonal polynomials and positivity,” pp. 64–85 in Special functions and wave propagation (Washington DC, 9–12 June 1969). Edited by D. Ludwig and F. W. J. Olver. Studies in Applied Mathematics 6. SIAM (Philadelphia), 1970. A version of this was published as a 1969 technical report. incollection

Donald A. Ludwig	Related
Frank William John Olver	Related

[35] R. Askey: “An inequality for the classical polynomials,” Nederl. Akad. Wet., Proc., Ser. A 73 (1970), pp. 22–25. MR 257428 Zbl 0186.37701 article

[36] R. Askey and R. P. Boas, Jr.: “Some integrability theorems for power series with positive coefficients,” pp. 23–32 in Mathematical essays dedicated to A. J. Macintyre. Edited by H. Shankar. Ohio University Press (Athens, OH), 1970. MR 277956 Zbl 0212.41401 incollection

Hari Shankar	Related
Ralph Philip Boas, Jr.	Related
Archibald James Macintyre	Related

[37] R. Askey: “Linearization of the product of orthogonal polynomials,” pp. 131–138 in Problems in analysis: A symposium in honor of Salomon Bochner (Princeton, NJ, 1–3 April 1969). Edited by R. C. Gunning. Princeton Mathematical Series 31. Princeton University Press, 1970. MR 344786 Zbl 0212.41001 incollection

Robert Clifford Gunning	Related
Salomon Bochner	Related

[38] R. Askey and S. Karlin: “Some elementary integrability theorems for special transforms,” J. Analyse Math. 23 (1970), pp. 27–38. MR 270078 Zbl 0212.41402 article

[39] R. Askey: Certain rational functions whose power series have positive coefficients. Technical report ZW 1970-006, Mathematisch Centrum, Amsterdam, 1970. This was later published as an article in Am. Math. Monthly 79:4 (1972). Zbl 0198.08402 techreport

[40] L. Carlitz: “A positive integral (Richard Askey),” SIAM Review 12 : 4 (1970), pp. 588–590. article

[41] R. Askey and G. Gasper: “Linearization of the product of Jacobi polynomials, III,” Can. J. Math. 23 : 2 (1971), pp. 332–338. MR 273273 Zbl 0212.40904 article

[42] R. Askey and G. Gasper: “Jacobi polynomial expansions of Jacobi polynomials with non-negative coefficients,” Proc. Cambridge Philos. Soc. 70 : 2 (1971), pp. 243–255. MR 296369 Zbl 0217.11402 article

[43] R. Askey: “Orthogonal expansions with positive coefficients, II,” SIAM J. Math. Anal. 2 : 3 (1971), pp. 340–346. Part I was published in Proc. Am. Math. Soc. 16:6 (1965). MR 296591 Zbl 0202.35302 article

[44] G. Gasper: “A positive integral (Richard Askey),” SIAM Review 13 : 3 (1971), pp. 396–397. article

[45] R. Askey: “Book review, P. L. Butzer and R. J. Nessel, ‘Fourier analysis and approximation’,” SIAM Rev. 14 : 3 (1972), pp. 507–508. article

Rolf Joachim Nessel	Related
Paul Leo Butzer	Related

[46] R. Askey and J. Fitch: “A positive Cesàro mean,” Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 381/409 (1972), pp. 131–134. MR 330902 Zbl 0254.42001 article

[47] R. Askey and G. Gasper: “Certain rational functions whose power series have positive coefficients,” Am. Math. Monthly 79 : 4 (April 1972), pp. 327–341. Part II was published in SIAM J. Math. Anal. 5:1 (1974). This also appeared as a 1970 technical report. MR 301441 Zbl 0242.33023 article

[48] R. Askey: “Positivity of the Cotes numbers for some Jacobi abscissas,” Numer. Math. 19 (1972), pp. 46–48. Part II appeared in J. Inst. Math. Appl. 24:1 (1979). MR 301897 Zbl 0237.65012 article

[49] R. Askey: “Mean convergence of orthogonal series and Lagrange interpolation,” Acta Math. Acad. Sci. Hungar. 23 : 1–2 (1972), pp. 71–85. This originally appeared as a 1969 technical report. MR 322400 Zbl 0253.41003 article

[50] R. Askey: “Jacobi summability,” J. Approximation Theory 5 : 4 (June 1972), pp. 387–392. Dedicated to Professor J. L. Walsh on the occasion of his 75th birthday. MR 336206 Zbl 0234.40016 article

[51] R. Askey: “Positive Jacobi polynomial sums,” Tohoku Math. J. (2) 24 : 2 (1972), pp. 109–119. Dedicated to Professor Gen-ichirô Sunouchi on his 60th birthday. Part II was published in Am. J. Math 98:3 (1976). Part III was published in Linear operators and approximation, II (1974). MR 340672 Zbl 0237.33010 article

[52] C. C. Rousseau, R. Askey, and G. Gasper: “Problem 71-13: ‘A positive integral’ by L. Carlitz,” SIAM Rev. 14 (1972), pp. 372–378. MR 304731 article

[53] R. Askey and B. Razban: “An integral for Jacobi polynomials,” Simon Stevin 46 (1972–1973), pp. 165–169. MR 328160 Zbl 0308.33006 article

[54] R. Askey: Radial characteristic functions. Technical report 1262, UW-Madison Mathematics Research Center, November 1973. techreport

[55] R. Askey: “Refinements of Abel summability for Jacobi series,” pp. 335–338 in Harmonic analysis on homogeneous spaces (Williamstown, MS, 31 July–18 August 1972). Edited by C. C. Moore. Proceedings of Symposia in Pure Mathematics 26. American Mathematical Society (Providence, RI), 1973. MR 340934 Zbl 0282.43008 incollection

[56] R. Askey: “Summability of Jacobi series,” Trans. Am. Math. Soc. 179 (1973), pp. 71–84. MR 315351 Zbl 0268.33015 article

[57] R. Askey: “Grünbaum’s inequality for Bessel functions,” J. Math. Anal. Appl. 41 : 1 (January 1973), pp. 122–124. MR 316769 Zbl 0253.33009 article

[58] R. Askey: “Positive Jacobi polynomial sums, III,” pp. 305–312 in Linear operators and approximation, II (Oberwolfach, Germany, 30 March–6 April 1974). Edited by P. L. Butzer and B. Szökefalvi-Nagy. International Series on Numerical Mathematics. Birkhäuser (Basel), 1974. Part I was published in Tohoku Math. J. 24:2 (1972). Part II was published in Am. J. Math 98:3 (1976). MR 430359 Zbl 0299.42001 incollection

Béla Szőkefalvi-Nagy	Related
Paul Leo Butzer	Related

[59] R. Askey: “Certain rational functions whose power series have positive coefficients, II,” SIAM J. Math. Anal. 5 : 1 (1974), pp. 53–57. Part I was published in Am. Math. Monthly 79:4 (1972). MR 333572 Zbl 0291.33010 article

[60] R. Askey and J. Steinig: “Some positive trigonometric sums,” Trans. Am. Math. Soc. 187 (1974), pp. 295–307. MR 338481 Zbl 0244.42002 article

Vietoris found an interesting generalization of the classical inequality \[ \sum_{k=1}^n\frac{\sin k\theta}k > 0 ,\] \( 0 < \theta < \pi \). A simplified proof is given for his inequality and his similar inequality for cosine series. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums of ultraspherical polynomials which extend an old inequality of Fejér. Both of Vietoris’ inequalities are special cases of a general problem for Jacobi polynomials, and a summary is given of known results on this problem.

[61] R. Askey and H. Pollard: “Some absolutely monotonic and completely monotonic functions,” SIAM J. Math. Anal. 5 : 1 (1974), pp. 58–63. MR 340935 Zbl 0239.26010 article

[62] R. Askey: “Positive Cesàro means of numerical series,” Proc. Am. Math. Soc. 45 : 1 (July 1974), pp. 63–68. MR 342900 Zbl 0288.40010 article

[63] R. Askey: “Some absolutely monotonic functions,” Studia Sci. Math. Hungar. 9 (1974), pp. 51–56. A possibly related brief note with the same title was published in Acta Math. Acad. Sci. Hungar. 33:1–2 (1979). MR 372518 Zbl 0298.26010 article

[64] R. Askey: “Jacobi polynomials, I: New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum,” SIAM J. Math. Anal. 5 : 1 (1974), pp. 119–124. MR 385197 Zbl 0242.33019 article

[65] A. A. Jagers: “An integral inequality (Richard Askey),” SIAM Review 16 : 2 (1974), pp. 262–263. article

[66] R. Askey: “One-sided approximation to special functions,” SIAM Review 16 : 4 (1974), pp. 545–546. article

[67] A. A. Jagers: “A sine inequality (R. A. Askey),” SIAM Review 16 : 4 (1974), pp. 550–553. article

[68] R. Askey: A note on the history of series. Technical report 1532, UW-Madison Mathematics Research Center, March 1975. techreport

[69] R. Askey: “Book reviews: F. Oberhettinger, ‘Tables of Bessel transforms’, ‘Fourier expansions: A collection of formulas’, ‘Fourier transforms of distributions and their inverses: A collection of tables’; F. Oberhettinger and L. Badii, ‘Tables of Laplace transforms’,” SIAM Rev. 17 : 4 (1975), pp. 707–712. article

[70] R. Askey, M. E. H. Ismail, and T. M. T. Rashed: A derangement problem. Technical report 1522, UW-Madison Mathematics Research Center, June 1975. Abstract also published in Proceedings of the sixth southeastern conference on combinatorics, graph theory, and computation (1975). techreport

Mourad El Houssieny Ismail	Related
Thanaa M. T. Rashed	Related

[71] R. Askey and M. E. H. Ismail: A combinatorial sum. Technical report 1557, UW-Madison Mathematics Research Center, July 1975. techreport

[72] Theory and application of special functions (Madison, WI, 21 March–2 April 1975). Edited by R. A. Askey. University of Wisconsin Mathematics Research Center Publications 35. Academic Press (New York), 1975. MR 379915 Zbl 0316.00014 book

[73] R. Askey: Orthogonal polynomials and special functions. CBMS-NSF Regional Conference Series in Applied Mathematics 21. SIAM (Philadelphia), 1975. MR 481145 Zbl 0298.33008 book

[74] R. Askey: “Inequalities via fractional integration,” pp. 106–115 in Fractional calculus and its applications (New Haven, CT, 15–16 June 1974). Edited by B. Ross. Lecture Notes in Mathematics 457. Springer (Berlin), 1975. MR 481146 Zbl 0307.26014 incollection

[75] R. Askey, G. Gasper, and M. E.-H. Ismail: “A positive sum from summability theory,” J. Approximation Theory 13 : 4 (April 1975), pp. 413–420. Dedicated to G. G. Lorentz on the occasion of his sixty-fifth birthday. MR 369978 Zbl 0295.40008 article

Mourad El Houssieny Ismail	Related
George Gunter Lorentz	Related
George Gasper, Jr.	Related

[76] R. Askey: “Some characteristic functions of unimodal distributions,” J. Math. Anal. Appl. 50 : 3 (June 1975), pp. 465–469. MR 370684 Zbl 0308.60016 article

[77] R. Askey, G. Gasper, and L. A. Harris: “An inequality for Tchebycheff polynomials and extensions,” J. Approximation Theory 14 : 1 (May 1975), pp. 1–11. MR 372285 Zbl 0297.33018 article

George Gasper, Jr.	Related
Lawrence Albert Harris	Related

[78] R. Askey, M. E. H. Ismail, and T. M. T. Rashed: “A derangement problem (abstract),” pp. 639–641 in Proceedings of the sixth southeastern conference on combinatorics, graph theory, and computation (Boca Raton, FL, 17–20 February 1975), vol. 14. Edited by F. Hoffman, R. C. Mullin, R. B. Levow, D. Roselle, R. G. Stanton, and R. S. D. Thomas. Congressus Numerantium. Utilitas Mathematica (Winnipeg), 1975. Also published as a 1975 technical report. Zbl 0337.05011 incollection

Frederick Mullin Hoffman	Related
Ronald C. Mullin	Related
David Paul Roselle	Related
Ralph Gordon Stanton	Related
Mourad El Houssieny Ismail	Related
Thanaa M. T. Rashed	Related
R. B. Levow	Related
Robert S. D. Thomas	Related

[79] A. R. Holt: “An integral inequality (R. Askey),” SIAM Review 17 : 4 (1975), pp. 691–693. article

[80] R. Askey and M. E. H. Ismail: “Permutation problems and special functions,” Can. J. Math. 28 : 4 (1976), pp. 853–874. MR 406808 Zbl 0313.05005 article

[81] R. Askey and J. Steinig: “A monotonic trigonometric sum,” Am. J. Math. 98 : 2 (1976), pp. 357–365. MR 407513 Zbl 0334.42003 article

[82] R. Askey and G. Gasper: “Positive Jacobi polynomial sums, II,” Am. J. Math. 98 : 3 (1976), pp. 709–737. Part I was published in Tohoku Math. J. 24:2 (1972). Part III was published in Linear operators and approximation, II (1974). MR 430358 Zbl 0355.33005 article

Among the positive polynomial sums of Jacobi polynomials, there are two which have been very useful, the Cesàro means of the formal reproducing kernel and the sum \[ \sum_{k=0}^n \frac{P_k^{(\alpha,\beta)}(x)}{P_k^{(\beta,\alpha)}(1)} , \tag{*} \] which was first considered by Fejér when \( \alpha = 1/2 \), \( \beta = \pm 1/2 \) and when \( \alpha = \beta = 0 \). A conjecture is given which connects these two sets of inequalities and this conjecture is proven for many values of \( (\alpha,\beta) \). In particular, it is shown that if \( \beta\geq 0 \), then the sum (*) is nonnegative for \( -1 \leq x \leq 1 \) if and only if \( \alpha + \beta \geq -2 \). It is also shown that \[ \sum_{k=0}^n \frac{(\lambda+1)_{n-k}}{(n-k)!} \frac{(\lambda+1)_k}{k!} \frac{\sin(k+1)\theta}{k+1} > 0 ,\] \( 0 < \theta < \pi \), \( -1 < \lambda \leq 1 \), and that for real \( \alpha \) the function \[ (1 - r)^{-|\alpha|}[1 \pm r + (1 - 2xr + r^2)^{1/2}]^{\alpha} \] is absolutely monotonic for \( -1 \leq x \leq 1 \), i.e., it has nonnegative power series coefficients when it is expanded in a power series in \( r \). Limiting cases involving Laguerre and Hermite polynomials are also considered.

[83] R. Askey: “Problem 74-19, “One-sided approximation to special functions’ ’, by R. Askey,” SIAM Rev. 18 : 1 (1976), pp. 121–122. MR 393598 article

[84] R. Askey: “One-sided approximation to special functions,” SIAM Review 18 : 1 (1976), pp. 121–122. article

[85] R. Askey and N. H. Bingham: “Gaussian processes on compact symmetric spaces,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 37 : 2 (1976–1977), pp. 127–143. MR 423000 Zbl 0329.60019 article

Paul Lévy studied Gaussian processes \( \xi(a) \) with the parameter a running over Euclidean \( d \)-space \( R^d \) and he also studied the case when a runs over the \( d \)-sphere \( S^d \). His results were extended by Gangolli in a number of directions, one being the extension to the cases where the parameter \( a \) lies in the other two-point homogeneous Riemannian manifolds. In the compact cases Gangolli showed there was a distinction between spheres and projective spaces, in that the process discovered by Lévy which he called Brownian motion parametrized by spheres does not exist for projective spaces. However many interesting Gaussian process exist with parameters running through projective spaces as we show.

[86] R. Askey: “Hypergeometric functions,” pp. 658–659 in Encyclopedia of science and technology, 4th edition, vol. 6. Edited by D. N. Lapedes. McGraw-Hill (New York), 1977. incollection

[87] R. Askey: “Legendre functions” in Encyclopedia of science and technology, 4th edition, vol. 7. Edited by D. N. Lapedes. McGraw-Hill (New York), 1977. incollection

[88] R. Askey: “Book review: Y. L. Luke: ‘Mathematical functions and their approximations’,” Math. Comput. 31 : 137 (January 1977), pp. 323. article

[89] R. Askey: “Special functions” in Encyclopedia of science and technology, 4th edition, vol. 12. Edited by D. N. Lapedes. McGraw-Hill (New York), 1977. incollection

[90] W. Miller, Jr.: Symmetry and separation of variables. Encyclopedia of Mathematics and its Applications 4. Addison-Wesley, 1977. With a foreword by Richard Askey. This was reprinted in 2011. MR 460751 Zbl 0368.35002 book

[91] R. Askey and D. T. Haimo: “Series inversion of some convolution transforms,” J. Math. Anal. Appl. 59 : 1 (1977), pp. 119–129. MR 442607 Zbl 0353.44005 article

[92] R. Askey and G. Gasper: “Convolution structures for Laguerre polynomials,” J. Analyse Math. 31 (1977), pp. 48–68. MR 486692 Zbl 0347.33006 article

[93] G. E. Andrews and R. Askey: “Enumeration of partitions: The role of Eulerian series and \( q \)-orthogonal polynomials,” pp. 3–26 in Higher combinatorics (West Berlin, 1–10 September 1976). Edited by M. Aigner. NATO Advanced Study Institutes Series: Series C, Mathematical and Physical Sciences 31. Reidel (Dordrecht), 1977. MR 519776 Zbl 0381.10008 incollection

The theory of partitions has long been associated with so called basic hypergeometric functions or Eulerian series. We begin with discussion of some of the lesser known identities of L. J. Rogers which have interesting interpretations in the theory of partitions. Illustrations are given for the numerous ways partition studies lead to Eulerian series. The main portion of our work is primarily an introduction to recent work on orthogonal polynomials defined by basic hypergeometric series and to the applications that can be made of these results to the theory of partitions. Perhaps it is most interesting to note that we deduce the Rogers–Ramanujan identities from our solution to the connection coefficient problem for the little \( q \)-Jacobi polynomials.

George W. Eyre Andrews	Related
Martin S. Aigner	Related

[94] R. Askey: “Book review, H. Exton, ‘Multiple hypergeometric functions and applications’,” SIAM Rev. 20 : 4 (1978), pp. 874–875. article

[95] A. G. Mackie: “Arthur Erdélyi,” Appl. Anal. 8 : 1 (1978), pp. 1–10. With addendum by Richard Askey, “Retrospective on the Bateman project” on pp. 5–10. MR 516518 Zbl 0393.01009 article

Arthur Erdélyi	Related
A. G. Mackie	Related

[96] R. Askey: “Jacobi’s generating function for Jacobi polynomials,” Proc. Am. Math. Soc. 71 : 2 (September 1978), pp. 243–246. MR 486693 Zbl 0393.33010 article

[97] R. Askey, M. E. H. Ismail, and T. Koornwinder: “Weighted permutation problems and Laguerre polynomials,” J. Combin. Theory Ser. A 25 : 3 (November 1978), pp. 277–287. MR 514623 Zbl 0405.05008 article

Mourad El Houssieny Ismail	Related
Tom Hendrik Koornwinder	Related

[98] G. E. Andrews and R. Askey: “A simple proof of Ramanujan’s summation of the \( {}_1 \psi_1 \),” Aequationes Math. 18 : 3 (1978), pp. 333–337. MR 522519 Zbl 0401.33002 article

[99] R. Askey: “The \( q \)-gamma and \( q \)-beta functions,” Applicable Anal. 8 : 2 (1978–1979), pp. 125–141. MR 523950 Zbl 0398.33001 article

[100] R. Askey: “Book review, B. C. Carlson, ‘Special functions of applied mathematics’,” SIAM Rev. 21 : 3 (1979), pp. 420–421. article

[101] R. Askey: “Book review, H. Exton, ‘Handbook of hypergeometric integrals: Theory, applications, tables, computer programs’,” SIAM Rev. 21 : 4 (1979), pp. 578. article

[102] R. Askey: “Book review: G. E. Andrews, ‘The theory of partitions’,” Bull. Am. Math. Soc. (N.S.) 1 : 1 (1979), pp. 203–210. MR 1567111 article

[103] R. Askey: “Some absolutely monotonic functions,” Acta Math. Acad. Sci. Hungar. 33 : 1–2 (1979), pp. 3–5. To G. Alexits on his 80th birthday. A possibly related short article with the same title was published in Studia Sci. Math. Hungar. 9 (1974). MR 515116 Zbl 0391.33009 article

[104] R. Askey: “Positivity of the Cotes numbers for some Jacobi abscissas, II,” J. Inst. Math. Appl. 24 : 1 (1979), pp. 95–98. Part I appeared in Numer. Math. 19 (1972). MR 539375 Zbl 0416.65016 article

[105] R. Askey and J. Wilson: “A set of orthogonal polynomials that generalize the Racah coefficients or \( 6-j \) symbols,” SIAM J. Math. Anal. 10 : 5 (1979), pp. 1008–1016. MR 541097 Zbl 0437.33014 article

[106] R. Askey and M. E. H. Ismail: “The very well poised \( {}_6\psi_6 \),” Proc. Am. Math. Soc. 77 : 2 (November 1979), pp. 218–222. MR 542088 Zbl 0387.33002 article

[107] R. Askey: “Comment on I. J. Good and R. A. Gaskins, ‘Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data’,” J. Am. Stat. Assoc. 75 : 369 (March 1980), pp. 69. article

[108] R. Askey: “Book review, A. M. Mathai and R. K. Saxena, ‘The \( H \) function with applications in statistics and other disciplines’,” SIAM Rev. 22 : 3 (1980), pp. 379–380. article

Arakaparampli Mathai Mathai	Related
Ram Kishore Saxena	Related

[109] R. Askey: “Ramanujan’s extensions of the gamma and beta functions,” Am. Math. Monthly 87 : 5 (May 1980), pp. 346–359. MR 567718 Zbl 0437.33001 article

[110] R. Askey: “Some basic hypergeometric extensions of integrals of Selberg and Andrews,” SIAM J. Math. Anal. 11 : 6 (1980), pp. 938–951. MR 595822 Zbl 0458.33002 article

[111] R. A. Askey and M. E. H. Ismail: “The Rogers \( q \)-ultraspherical polynomials,” pp. 175–182 in Approximation theory, III (Austin, TX, 8–12 January 1980). Edited by E. W. Cheney. Academic Press (New York), 1980. MR 602713 Zbl 0479.33013 incollection

Mourad El Houssieny Ismail	Related
George Gunter Lorentz	Related
Elliott Ward Cheney, Jr	Related

[112] R. Askey: “Some extensions of the gamma and beta functions,” Real Anal. Exch. 5 : 1 (1980), pp. 128–131. Zbl 0427.33001 article

[113] G. E. Andrews, R. Askey, and H. Flanders: “Problems and solutions: Solutions of problems dedicated to Emory P. Starke: S7,” Amer. Math. Monthly 87 : 5 (1980), pp. 403–404. MR 1539387 article

[114] G. E. Andrews and R. Askey: “Another \( q \)-extension of the beta function,” Proc. Am. Math. Soc. 81 : 1 (January 1981), pp. 97–100. MR 589145 Zbl 0471.33001 article

[115] R. Askey: “A \( q \)-extension of Cauchy’s form of the beta integral,” Quart. J. Math. Oxford Ser. (2) 32 : 3 (September 1981), pp. 255–266. MR 625639 Zbl 0463.33003 article

[116] G. Szegő: Collected papers, vol. 1: 1915-1927. Edited by R. Askey. Birkhäuser (Boston), 1982. MR 674482 Zbl 0491.01016 book

[117] R. Askey: “Book review: L. Lewin, ‘Polylogarithms and associated functions’,” Bull. Am. Math. Soc. (N.S.) 6 : 2 (1982), pp. 248–251. MR 1567360 article

[118] R. Askey, I. J. Schoenberg, and A. Sharma: “Hausdorff’s moment problem and expansions in Legendre polynomials,” J. Math. Anal. Appl. 86 : 1 (March 1982), pp. 237–245. MR 649868 Zbl 0483.44012 article

Ambikeshwar Sharma	Related
Isaac Jacob Schoenberg	Related

[119] R. Askey: “Two integrals of Ramanujan,” Proc. Am. Math. Soc. 85 : 2 (1982), pp. 192–194. MR 652440 Zbl 0503.33001 article

[120] R. Askey and J. Wilson: “A set of hypergeometric orthogonal polynomials,” SIAM J. Math. Anal. 13 : 4 (1982), pp. 651–655. MR 661596 Zbl 0496.33007 article

[121] R. Askey: “An integral of products of Legendre functions and a Clebsch–Gordan sum,” Lett. Math. Phys. 6 : 4 (1982), pp. 299–302. MR 668464 Zbl 0524.33004 article

[122] R. Askey: “A \( q \)-beta integral associated with \( BC_1 \),” SIAM J. Math. Anal. 13 : 6 (1982), pp. 1008–1010. MR 674769 Zbl 0501.33002 article

[123] G. Szegő: Collected papers, vol. 2: 1927–1943. Edited by R. Askey. Birkhäuser (Boston), 1982. MR 674483 book

[124] G. Szegő: Collected papers, vol. 3: 1945–1972. Edited by R. Askey. Birkhäuser (Boston), 1982. MR 674484 book

[125] R. Askey: “An elementary evaluation of a beta type integral,” Indian J. Pure Appl. Math. 14 : 7 (1983), pp. 892–895. MR 714840 Zbl 0523.33001 article

[126] R. Askey and M. E. H. Ismail: “A generalization of ultraspherical polynomials,” pp. 55–78 in Studies in pure mathematics: To the memory of Paul Turán. Edited by P. Erdős. Birkhäuser (Basel), 1983. MR 820210 Zbl 0532.33006 incollection

Mourad El Houssieny Ismail	Related
Pál (Paul) Turán	Related
Paul Erdős	Related

[127] R. Askey, P. S. Bruckman, and P. Erdos: “Problems and solutions: Advanced problems: 6445–6447,” Amer. Math. Monthly 90 : 10 (1983), pp. 709–710. MR 1540315 article

[128] R. Askey: “A \( q \)-extension of Cauchy’s beta integral,” SIAM Rev 26 : 2 (April 1984), pp. 270. article

[129] Special functions: Group theoretical aspects and applications. Edited by R. A. Askey, T. H. Koornwinder, and W. Schempp. Mathematics and Its Applications 18. Reidel (Dordrecht), 1984. MR 774053 Zbl 0543.00007 book

Tom Hendrik Koornwinder	Related
Walter Johannes Schempp	Related

[130] R. Askey and J. A. Wilson: “A recurrence relation generalizing those of Apéry,” J. Austral. Math. Soc. Ser. A 36 : 2 (1984), pp. 267–278. MR 725750 Zbl 0558.33003 article

[131] R. Askey: “The very well poised \( {}_6\psi_{6} \), II,” Proc. Am. Math. Soc. 90 : 4 (April 1984), pp. 575–579. MR 733409 Zbl 0509.33001 article

[132] R. Askey and J. Wimp: “Associated Laguerre and Hermite polynomials,” Proc. Roy. Soc. Edinburgh Sect. A 96 : 1–2 (1984), pp. 15–37. MR 741641 Zbl 0547.33006 article

[133] W. Al-Salam, W. R. Allaway, and R. Askey: “Sieved ultraspherical polynomials,” Trans. Am. Math. Soc. 284 : 1 (1984), pp. 39–55. MR 742411 Zbl 0547.33005 article

The continuous \( q \)-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway’s Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway’s and showed that these polynomials arose from the \( q \)-ultraspherical polynomials when \( q \) approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in \( (-1,1) \). Generating functions and other formulas are also found.

William R. Allaway	Related
Waleed A. Al-Salam	Related

[134] R. Askey and M. Ismail: Recurrence relations, continued fractions, and orthogonal polynomials. Memoirs of the American Mathematical Society 300. American Mathematical Society (Providence, RI), 1984. MR 743545 Zbl 0548.33001 book

[135] R. Askey: “Remarks on the preceding paper by Gavin Brown and Edwin Hewitt,” Math. Ann. 268 : 1 (1984), pp. 123–126. MR 744331 Zbl 0553.42001 article

Edwin Hewitt	Related
Gavin Brown	Related

[136] W. Al-Salam, W. R. Allaway, and R. Askey: “A characterization of the continuous \( q \)-ultraspherical polynomials,” Can. Math. Bull. 27 : 3 (1984), pp. 329–336. MR 749641 Zbl 0537.33004 article

William R. Allaway	Related
Waleed A. Al-Salam	Related

[137] R. Askey and A. Regev: “Maximal degrees for Young diagrams in a strip,” European J. Combin. 5 : 3 (September 1984), pp. 189–191. MR 765624 Zbl 0555.05009 article

[138] R. Askey: “Preface,” pp. xi–xxxiv in Special functions: Group theoretical aspects and applications. Edited by R. A. Askey, T. H. Koornwinder, and W. Schempp. Mathematics and Its Applications 18. Reidel (Dordrecht), 1984. MR 774054 incollection

Tom Hendrik Koornwinder	Related
Walter Johannes Schempp	Related

[139] R. Askey: “Orthogonal polynomials old and new, and some combinatorial connections,” pp. 67–84 in Enumeration and design (Waterloo, ON, 14 June–2 July 1982). Edited by D. M. Jackson and S. A. Vanstone. Academic Press (Toronto), 1984. MR 782309 Zbl 0555.05008 incollection

David M. R. Jackson	Related
Scott Alexander Vanstone	Related

[140] R. Askey: “Orthogonal polynomials and some definite integrals,” pp. 935–943 in Proceedings of the International Congress of Mathematicians (Warsaw, 16–24 August 1983), vol. 2. Edited by Z. Ciesielski and C. Olech. PWN (Warsaw), 1984. MR 804747 Zbl 0562.33007 incollection

Czesław Olech	Related
Zbigniew Ciesielski	Related

[141] C. L. Mallows, G. Emmanuele, A. Villani, and R. Askey: “Problems and solutions: Advanced problems: 6451–6453,” Amer. Math. Monthly 91 : 2 (1984), pp. 144. MR 1540360 article

[142] R. Askey: “Basic hypergeometric extensions of some integrals,” SIAM Rev 27 : 4 (December 1985), pp. 574. article

[143] R. Askey: “Book review, T. S. Chihara, ‘An introduction to orthogonal polynomials’,” J. Approx. Theory 43 : 4 (April 1985), pp. 394–395. article

[144] R. Askey and J. Wilson: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Memoirs of the American Mathematical Society 319. American Mathematical Society (Providence, RI), 1985. MR 783216 Zbl 0572.33012 book

[145] R. Askey: “Continuous Hahn polynomials,” J. Phys. A 18 : 16 (1985), pp. L1017–L1019. MR 812420 Zbl 0582.33007 article

[146] R. Askey: “Appendix,” pp. 311–318 in Number theory (Edmonton, AB, 28–30 April 1983), published as Rocky Mountain J. Math. 15 : 2. Issue edited by M. V. Subbarao. Arizona State University (Tempe, AZ), 1985. MR 823242 Zbl 0589.33003 incollection

[147] G. E. Andrews and R. Askey: “Classical orthogonal polynomials,” pp. 36–62 in Polynômes orthogonaux et applications [Orthogonal polynomials and applications] (Bar-le-Duc, France, 15–18 October 1984). Edited by C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux. Lecture Notes in Mathematics 1171. Springer (Berlin), 1985. MR 838970 Zbl 0596.33016 incollection

André Draux	Related
Pascal Maroni	Related
André Ronveaux	Related
Alphonse Ph. Magnus	Related
George W. Eyre Andrews	Related
Claude Brezinski	Related

[148] R. Askey: “Some problems about special functions and computations,” pp. 1–22 in Convegno su “Special functions: Theory and computation” [Conference on “Special functions: Theory and computation”] (Turin, Italy, 10–12 October 1984), published as Rend. Sem. Mat. Univ. Politec. Torino Special Issue. Issue edited by G. Allasia, L. Gatteschi, W. Gautschi, and G. Monegato. Universitá degli Studi di Torino, 1985. MR 850027 Zbl 0596.33001 incollection

Giovanni Monegato	Related
Luigi Gatteschi	Related
Walter Gautschi	Related
Giampietro Allasia	Related

[149] R. Askey and M. E. H. Ismail: “Problems and solutions: Solutions of advanced problems: 6445,” Amer. Math. Monthly 92 : 4 (1985), pp. 294. MR 1540629 article

[150] R. Szwarc and R. Askey: “Problems and solutions: Advanced problems: 6496–6497,” Amer. Math. Monthly 92 : 5 (1985), pp. 362–363. MR 1540649 article

[151] R. Askey: “Problems and solutions: Solutions of advanced problems: 6453,” Amer. Math. Monthly 92 : 7 (1985), pp. 516. MR 1540698 article

[152] J. A. Wilson: “A \( q \)-extension of Cauchy’s beta integral (Richard Askey),” SIAM Review 27 : 2 (1985), pp. 252–253. article

[153] R. Askey: “On the sign of a set of hypergeometric functions,” SIAM Rev 28 : 1 (March 1986), pp. 86. article

[154] R. Askey, T. H. Koornwinder, and M. Rahman: “An integral of products of ultraspherical functions and a \( q \)-extension,” J. London Math. Soc. (2) 33 : 1 (1986), pp. 133–148. MR 829394 Zbl 0564.33008 article

Let \( P_n(x) \) and \( Q_n(x) \) denote the Legendre polynomial of degree \( n \) and the usual second solution to the differential equation, respectively. Din showed that \[ \int_{-1}^1 Q_n(x)P_m(x)P_l(x)\,dx \] vanishes when \( |l-m| < n < l + m \), and Askey evaluated the integral for arbitrary integral values of \( l \), \( m \) and \( n \). We extend this to the evaluation of \[ \int_{-1}^1 D_n^{\lambda}(x)C_m^{\lambda}(x)C_l^{\lambda}(x)(1 - x^2)^{2\lambda - 1}\,dx ,\] where \( C_n^{\lambda}(x) \) is the ultraspherical polynomial and \( C_n^{\lambda}(x) \) is the appropriate second solution to the ultraspherical differential equation. A \( q \)-extension is found using the continuous \( q \)-ultraspherical polynomials of Rogers. Again the integral vanishes when \( |l - m| < n < l + m \). It is shown that this vanishing phenomenon holds for quite general orthogonal polynomials. A related integral of the product of three Bessel functions is also evaluated.

Tom Hendrik Koornwinder	Related
Mizan Rahman	Related

[155] R. Askey: “Limits of some \( q \)-Laguerre polynomials,” J. Approx. Theory 46 : 3 (March 1986), pp. 213–216. MR 840392 Zbl 0641.33018 article

[156] R. Askey and R. Roy: “More \( q \)-beta integrals,” Rocky Mountain J. Math. 16 : 2 (1986), pp. 365–372. MR 843057 Zbl 0599.33002 article

[157] R. Askey and G. Gasper: “Inequalities for polynomials,” pp. 7–32 in The Bieberbach conjecture (West Lafayette, IN, 11–14 March 1985). Edited by D. Drasin, P. Duren, and A. Marden. Mathematical Surveys and Monographs 21. American Mathematical Society (Providence, RI), 1986. MR 875228 incollection

Peter L. Duren	Related
David Drasin	Related
Albert Marden	Related
George Gasper, Jr.	Related

[158] R. Askey: “My reaction to de Branges’s proof of the Bieberbach conjecture,” pp. 213–215 in The Bieberbach conjecture (West Lafayette, IN, 11–14 March 1985). Edited by D. Drasin, P. Duren, and A. Marden. Mathematical Surveys and Monographs 21. American Mathematical Society (Providence, RI), 1986. MR 875244 incollection

Peter L. Duren	Related
Louis de Branges	Related
David Drasin	Related
Albert Marden	Related

[159] R. Askey: “Positive quadrature methods and positive polynomial sums,” pp. 1–29 in Approximation theory, V (Boston). Edited by C. K. Chui, L. L. Schumaker, and J. D. Ward. Academic Press, 1986. MR 903680 Zbl 0613.41027 incollection

Charles Kam-tai Chui	Related
Larry Lee Schumaker	Related
Joseph Dinneen Ward	Related

[160] R. Askey, R. E. Shafer, E. Kronheimer, and A. Lupas: “Problems and solutions: Advanced problems: 6514–6517,” Amer. Math. Monthly 93 : 4 (1986), pp. 304–305. MR 1540850 article

Erwin H. Kronheimer	Related
Alexandru Ioan Lupas	Related

[161] R. Askey: “On the sign of a set of hypergeometric functions,” SIAM Review 28 : 1 (1986), pp. 86. article

[162] R. Askey: “Ramanujan,” Math. Intell. 9 : 1 (March 1987), pp. 4–5. MR 869531 article

[163] R. Askey: “Ramanujan’s \( {}_1\psi_1 \) and formal Laurent series,” Indian J. Math. 29 : 2 (1987), pp. 101–105. MR 919888 Zbl 0646.33002 article

[164] R. Askey: “An integral of Ramanujan and orthogonal polynomials,” J. Indian Math. Soc. (N.S.) 51 : 1–2 (1987), pp. 27–36. MR 988306 Zbl 0665.33001 article

[165] R. Askey and G. Patruno: “Problems and solutions: Solutions of advanced problems: 6514,” Amer. Math. Monthly 94 : 10 (1987), pp. 1012–1014. MR 1541225 article

[166] R. Askey and S. L. Damjanović: “On the sign of a set of hypergeometric functions (Richard Askey),” SIAM Review 29 : 1 (1987), pp. 129–131. article

[167] R. Askey: “Book review: Jonathan M. Borwein and Peter B. Borwein, ‘Pi and the AGM’,” Am. Math. Monthly 95 : 9 (November 1988), pp. 895. article

Peter Benjamin Borwein	Related
Jonathan Michael Borwein	Related

[168] Ramanujan revisited (Urbana-Champaign, IL, 1–5 June 1987). Edited by G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin. Academic Press (Boston), 1988. MR 938955 Zbl 0635.00003 book

Robert Alexander Rankin	Related
Kollagunta Gopalaiyer Ramanathan	Related
Bruce C. Berndt	Related
Srinivasa Iyengar Ramanujan	Related
George W. Eyre Andrews	Related

[169] R. Askey: “Book review: S. Ramanujan, ‘The lost notebook and other unpublished papers’,” Bull. Am. Math. Soc. (N.S.) 19 : 2 (1988), pp. 558–560. MR 1567721 article

[170] R. Askey: “Beta integrals in Ramanujan’s papers, his unpublished work and further examples,” pp. 561–590 in Ramanujan revisited (Urbana-Champaign, IL, 1–5 June 1987). Edited by G. E. Andrews, R. A. Askey, B. C. Berndt, K. G. Ramanathan, and R. A. Rankin. Academic Press (Boston), 1988. MR 938979 Zbl 0648.33001 incollection

Robert Alexander Rankin	Related
Kollagunta Gopalaiyer Ramanathan	Related
Bruce C. Berndt	Related
Srinivasa Iyengar Ramanujan	Related
George W. Eyre Andrews	Related

[171] R. Askey: “How can mathematicians and mathematical historians help each other?,” pp. 201–217 in History and philosophy of modern mathematics (Minneapolis, MN, 17–19 May 1985). Edited by W. Aspray and P. Kitcher. Minnesota Studies in the Philosophy of Science 11. University of Minnesota Press (Minneapolis), 1988. MR 945472 Zbl 1366.01003 incollection

Philip Stuart Kitcher	Related
William Aspray	Related

[172] R. Askey: “Ramanujan and important formulae,” pp. 32–41 in Srinivasa Ramanujan (1887–1920): A Tribute. Edited by K. R. Nagarajan and T. Soundararajan. Macmillan India (Madras), 1988. MR 951693 incollection

T. Soundararajan	Related
Srinivasa Iyengar Ramanujan	Related
Kolumum R. Nagarajan	Related

[173] R. Askey: “Beta integrals and \( q \)-extensions,” pp. 85–102 in Proceedings of the Ramanujan centennial international conference (Annamalainagar, India, 15–18 December 1987). Edited by R. Balakrishnan, K. S. Padmanabhan, and V. Thangaraj. RMS Publications 1. Ramanujan Mathematical Society (Trichy, India), 1988. MR 993347 Zbl 0697.33002 incollection

Karaikurichi Srinivasan Padmanabhan	Related
V. Thangaraj	Related
Srinivasa Iyengar Ramanujan	Related
Rangaswami Balakrishnan	Related

[174] R. Askey: “Computer algebra and definite integrals,” pp. 121–128 in Computer algebra (New York, 5–6 April 1984). Edited by D. V. Chudnovsky and R. D. Jenks. Lecture Notes in Pure and Applied Mathematics 113. Dekker (New York), 1989. MR 1002980 Zbl 0672.33001 incollection

David Volfovich Chudnovsky	Related
Richard Dimick Jenks	Related

[175] A century of mathematics in America, Part 2. Edited by P. Duren. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. Edited with the assistance of Richard A. Askey and Uta C. Merzbach. MR 1003117 Zbl 0659.00007 book

Peter L. Duren	Related
Uta Caecilia Merzbach	Related

[176] R. Askey: “Orthogonal polynomials and theta functions,” pp. 299–321 in Theta functions: Bowdoin 1987 (Brunswick, ME, 6–24 July 1987), Part 2. Edited by L. Ehrenpreis and R. C. Gunning. Proceedings of Symposia in Pure Mathematics 49. American Mathematical Society (Providence, RI), 1989. MR 1013179 Zbl 0675.33006 incollection

Leon Ehrenpreis	Related
Robert Clifford Gunning	Related

[177] R. Askey: “Divided difference operators and classical orthogonal polynomials,” pp. 33–37 in Constructive Function theory — 86 Conference (Edmonton, AB, 22–26 July 1986), published as Rocky Mountain J. Math. 19 : 1. Issue edited by W. A. Al-Salam, Z. Ditzian, and S. D. Riemenschneider. Arizona State University (Tempe, AZ), 1989. MR 1016157 Zbl 0696.33008 incollection

Sherman Delbert Riemenschneider	Related
Waleed A. Al-Salam	Related
Zeev Ditzian	Related

[178] R. Askey: “Beta integrals and the associated orthogonal polynomials,” pp. 84–121 in Number theory (Madras, India, 21 December 1987). Edited by K. Alladi. Lecture Notes in Mathematics 1395. Springer (Berlin), 1989. MR 1019328 Zbl 0683.33002 incollection

[179] R. Askey: “Continuous \( q \)-Hermite polynomials when \( q > 1 \),” pp. 151–158 in \( q \)-series and partitions (Minneapolis, MN, 7–11 March 1988). Edited by D. Stanton. IMA Volumes in Mathematics and its Applications 18. Springer (New York), 1989. MR 1019849 Zbl 0694.33006 incollection

[180] A century of mathematics in America, Part 3. Edited by P. Duren. History of Mathematics 3. American Mathematical Society (Providence, RI), 1989. Edited with the assistance of Richard A. Askey, Harold M. Edwards and Uta C. Merzbach. MR 1025339 Zbl 0682.01001 book

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Harold Mortimer Edwards, Jr.	Related

[181] R. Askey: “Handbooks of special functions,” pp. 369–391 in A century of mathematics in America, Part 3. Edited by P. Duren, H. M. Edwards, and U. C. Merzbach. History of Mathematics 3. American Mathematical Society (Providence, RI), 1989. MR 1025353 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Harold Mortimer Edwards, Jr.	Related

[182] R. Askey and D. Richards: “Selberg’s second beta integral and an integral of Mehta,” pp. 27–39 in Probability, statistics, and mathematics: Papers in honor of Samuel Karlin. Edited by T. W. Anderson, K. B. Athreya, and D. L. Iglehart. Academic Press (Boston), 1989. MR 1031276 Zbl 0683.33001 incollection

Theodore W. Anderson	Related
Donald L. Iglehart	Related
Samuel Karlin	Related
Krishna B. Athreya	Related
Donald St. P. Richards	Related

[183] R. Askey: “Ramanujan and hypergeometric and basic hypergeometric series,” pp. 1–83 in Ramanujan international symposium on analysis. Edited by N. K. Thakare, K. C. Sharma, and T. T. Raghunathan. Macmillan India (New Delhi), 1989. Republished (in Russian) in Usp. Mat. Nauk 45:1(271) (1990), and (in English) in Russ. Math. Surv. 45:1 (1990) and Ramanujan: {Essays and surveys} (2001). MR 1117465 incollection

Krishna Chandra Sharma	Related
T. T. Raghunathan	Related
Nimbakrishna K. Thakare	Related
Srinivasa Iyengar Ramanujan	Related

[184] R. Askey: “Variants of Clausen’s formula for the square of a special \( {}_2F_1 \),” pp. 1–12 in Number theory and related topics: Papers presented at the Ramanujan colloquium (Bombay, 4–11 January 1988). Studies in Mathematics 12. Tata Institute of Fundamental Research (Bombay), 1989. MR 1441321 Zbl 0756.33002 incollection

[185] R. Askey: “Book review: N. J. Fine, ‘Basic hypergeometric series and applications’,” Bull. Am. Math. Soc. (N.S.) 21 : 1 (1989), pp. 130–132. MR 1567778 article

[186] C. W. Helstrom, O. G. Ruehr, and R. A. Askey: “Problems and solutions: Solution of advanced problems: 6541,” Amer. Math. Monthly 96 : 7 (1989), pp. 653–655. MR 1541569 article

[187] R. Askey: “Ramanujan and hypergeometric and basic hypergeometric series,” Russ. Math. Surv. 45 : 1 (1990), pp. 37–86. English version of Russian language article published in Usp. Mat. Nauk 45:1(271) (1990). This earlier appeared in Ramanujan international symposium on analysis (1989). Zbl 0722.33009 article

[188] R. Askey: “Relative extrema of Legendre functions of the second kind,” Proc. Indian Acad. Sci. Math. Sci. 100 : 1 (1990), pp. 1–8. MR 1051084 Zbl 0698.33006 article

[189] G. Gasper and M. Rahman: Basic hypergeometric series. Encyclopedia of Mathematics and its Applications 35. Cambridge University Press, 1990. With a foreword by Richard Askey. A Russian translation was published in 1993. 2nd edition published in 2004. MR 1052153 Zbl 0695.33001 book

Mizan Rahman	Related
George Gasper, Jr.	Related

[190] R. A. Askey: “Graphs as an aid to understanding special functions,” pp. 3–33 in Asymptotic and computational analysis (Winnipeg, MB, 5–7 June 1989). Edited by R. Wong. Lecture Notes in Pure and Applied Mathematics 124. Dekker (New York), 1990. MR 1052427 Zbl 0694.33002 incollection

[191] R. Askey: “Integration and computers,” pp. 35–82 in Computers in mathematics (Stanford, CA, 29 August–1 September 1986). Edited by D. V. Chudnovsky and R. D. Jenks. Lecture Notes in Pure and Applied Mathematics 125. Dekker (New York), 1990. MR 1068534 Zbl 0705.65012 incollection

David Volfovich Chudnovsky	Related
Richard Dimick Jenks	Related

[192] R. Askey: “The number of representations of an integer as the sum of two squares,” Indian J. Math. 32 : 2 (1990), pp. 187–192. MR 1069235 Zbl 0705.11019 article

[193] R. Askey and C. de Boor: “In memoriam: I. J. Schoenberg (1903–1990),” J. Approx. Theory 63 : 1 (October 1990), pp. 1–2. MR 1074077 Zbl 0703.01012 article

Carl-Wilhelm Reinhold de Boor	Related
Isaac Jacob Schoenberg	Related

[194] G. Gasper and M. Rahman: Basic hypergeometric series, 2nd edition. Encyclopedia of Mathematics and its Applications 96. Cambridge University Press, 1990. With a foreword by Richard Askey. Republication of 1990 original. MR 2128719 Zbl 1129.33005 book

Mizan Rahman	Related
George Gasper, Jr.	Related

[195] R. Askey: “Some comments on education,” Notices Am. Math. Soc. 39 : 5 (May–June 1992), pp. 424–426. article

[196] R. Askey: “Book review: K. Kodaira, ed., ‘Japanese grade 7–9 mathematics’,” College Math. J. 23 : 5 (November 1992), pp. 445–448. article

[197] R. Askey: “Book review: James B. Seaborn, ‘Hypergeometric functions and their applications’,” SIAM Rev 34 : 3 (September 1992), pp. 503. article

[198] R. Askey: “In memoriam: Antoni Zygmund (December 26, 1900–May 30, 1992),” J. Approx. Theory 71 : 1 (October 1992), pp. iii, 1–2. MR 1180870 Zbl 0753.01016 article

[199] R. Askey: “The world of \( q \),” CWI Q. 5 : 4 (1992), pp. 251–269. MR 1213742 Zbl 0765.33015 article

[200] R. Askey: “The Japanese university entrance examinations in mathematics,” MAA FOCUS 13 : 1 (February 1993), pp. 14. article

[201] R. Askey and S. K. Suslov: “The \( q \)-harmonic oscillator and an analogue of the Charlier polynomials,” J. Phys. A 26 : 15 (1993), pp. L693–L698. MR 1239353 Zbl 0859.33021 ArXiv math/9307206 article

[202] R. Askey and S. K. Suslov: “The \( q \)-harmonic oscillator and the Al-Salam and Carlitz polynomials,” Lett. Math. Phys. 29 : 2 (1993), pp. 123–132. MR 1245186 Zbl 0919.33010 ArXiv math/9307207 article

[203] R. Askey: “Problems which interest and/or annoy me,” pp. 3–15 in Proceedings of the seventh Spanish symposium on orthogonal polynomials and applications (Granada, Spain, 23–27 September 1991), published as J. Comput. Appl. Math. 48 : 1–2. Issue edited by J. S. Dehesa. Elsevier (Amsterdam), 1993. MR 1246848 Zbl 0797.33006 incollection

[204] R. Askey, N. M. Atakishiyev, and S. K. Suslov: “An analog of the Fourier transformation for a \( q \)-harmonic oscillator,” pp. 57–63 in Symmetries in science, VI: From the rotation group to quantum algebras (Bregenz, Austria, 2–7 August 1992). Edited by B. Gruber. Plenum (New York), 1993. MR 1258093 Zbl 0863.33020 ArXiv math/9307205 incollection

Sergei K. Suslov	Related
Bruno J. Gruber	Related
Natig M. Atakishiyev	Related

[205] D. Gasper and M. Rakhman: Bazisnye gipergeometricheskie ryady [Basic hypergeometric series]. Mir (Moscow), 1993. With a foreword by Richard Askey. Translated from the English and with a preface by N. M. Atakishiev and S. K. Suslov. Russian translation of 1990 original. MR 1286735 Zbl 0848.33011 book

Sergei K. Suslov	Related
Mizan Rahman	Related
George Gasper, Jr.	Related
Natig M. Atakishiyev	Related

[206] R. Askey and P. Névai: “Gábor Szegő: 1895–1985,” pp. 13–32 in Szegő Gábor emlékülés (Kunhegyes, Hungary, 21 January 1995), published as Mat. Lapok (N.S.) 3 : 3. Issue edited by P. György. Bolyai János Matematikai Társulat (Budapest), 1993. Hungarian translation of English original published in Math. Intell. 18:3 (1996). MR 1472807 incollection

Gábor Szegö	Related
Paul G. Névai	Related
Petruska György	Related

[207] R. Askey: “Quantum: An under-used resource,” Wisconsin Teacher Math. (1994), pp. 14–15. article

[208] R. Askey: “Gaussian quadrature in Ramanujan’s second notebook,” pp. 237–243 in K. G. Ramanathan memorial issue, published as Proc. Indian Acad. Sci. Math. Sci. 104 : 1. Springer India (New Dehli), February 1994. MR 1280071 Zbl 0794.41022 incollection

Srinivasa Iyengar Ramanujan	Related
Kollagunta Gopalaiyer Ramanathan	Related

[209] R. Askey: “Finite differences and orthogonal polynomials,” Rend. Mat. Appl. (7) 14 : 1 (1994), pp. 135–144. In memory of Professor A. Ghizzetti. MR 1284907 Zbl 0801.33005 article

[210] R. Askey: “A look at the Bateman project,” pp. 29–43 in The mathematical legacy of Wilhelm Magnus: Groups, geometry and special functions (Brooklyn, NY, 1–3 May 1992). Edited by W. Abikoff, J. S. Birman, and K. Kuiken. Contemporary Mathematics 169. American Mathematical Society (Providence, RI), 1994. MR 1292896 Zbl 0837.33001 incollection

William Abikoff	Related
Kathryn F. Kuiken	Related
Wilhelm Magnus	Related
Joan S. Birman	Related	Volume

[211] R. Askey: “Cube root algorithms,” Math. School 24 : 2 (March 1995), pp. 42–43. article

[212] R. Askey, N. M. Atakishiyev, and S. K. Suslov: “Fourier transformations for difference analogs of the harmonic oscillator,” pp. 140–144 in Problems on workshop on high energy physics and field theory (Protvino, Russia, 6–10 July 1992). Edited by A. P. Samokhin and G. L. Rcheulishvili. IHEP (Protvino, Russia), 1995. incollection

Sergei K. Suslov	Related
G. L. Rcheulishvili	Related
A. P. Samokhin	Related
Natig M. Atakishiyev	Related

[213] R. Askey: “Book review: I. M. Gelfand and A. Shen, ‘Algebra’,” Am. Math. Monthly 102 : 1 (January 1995), pp. 78–81. MR 1542635 article

Alexander Kh. Shen	Related
Israïl Moiseevich Gelfand	Related

[214] R. Askey: “Lars Onsager’s thesis,” pp. 921–922 in The collected works of Lars Onsager. World Scientific Series in 20th Century Physics 17. World Scientific (Singapore), 1996. incollection

[215] R. Askey and D. T. Haimo: “Similarities between Fourier and power series,” Am. Math. Monthly 103 : 4 (April 1996), pp. 297–304. In memory of Ralph P. Boas, Jr. MR 1383666 Zbl 0854.33005 article

Deborah Tepper Haimo	Related
Ralph Philip Boas, Jr.	Related

[216] R. Askey: “Dedication: Remembering Paul Turán,” J. Approx. Theory 86 : 3 (September 1996), pp. 253–254. MR 1405978 Zbl 0856.01047 article

[217] R. Askey and P. Nevai: “Gabor Szegő: 1895–1985,” Math. Intelligencer 18 : 3 (1996), pp. 10–22. A Hungarian translation was published in Mat. Lapok 3:3 (1993). MR 1412988 Zbl 0873.01030 article

Gábor Szegö	Related
Paul G. Névai	Related

[218] R. Askey: “A brief introduction to the world of \( q \),” pp. 13–20 in Symmetries and integrability of difference equations (Estérel, Canada, 22–29 May 1994). Edited by D. Levi, L. Vinet, and P. Winternitz. CRM Proceedings & Lecture Notes 9. American Mathematical Society (Providence, RI), 1996. MR 1416822 Zbl 0885.33010 incollection

Pavel Winternitz	Related
Luc Vinet	Related
Decio Levi	Related

[219] R. A. Askey, M. Rahman, and S. K. Suslov: “On a general \( q \)-Fourier transformation with nonsymmetric kernels,” J. Comput. Appl. Math. 68 : 1–2 (April 1996), pp. 25–55. MR 1418749 Zbl 0871.33008 ArXiv math/9411229 article

Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a \( q \)-Fourier transform by using the continuous \( q \)-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey–Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of \( q \)-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey–Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.

Sergei K. Suslov	Related
Mizan Rahman	Related

[220] R. Askey: “Chandra and Ramanujan,” pp. 130–138 in S. Chandrasekhar: The man behind the legend. Edited by K. C. Wali. World Scientific (River Edge, NJ), 1997. incollection

Kameshwar C. Wali	Related
Subramanyan Chandrasekhar	Related
Srinivasa Iyengar Ramanujan	Related

[221] R. Askey: “A new look at some old trigonometric expansions,” pp. 73–84 in Harmonic analysis and nonlinear differential equations (Riverside, CA, 3–5 November 1995). Edited by M. L. Lapidus, L. H. Harper, and A. J. Rumbos. Contemporary Mathematics 208. American Mathematical Society (Providence, RI), 1997. In honor of Victor Shapiro on his 70th birthday. MR 1467002 Zbl 0891.42012 incollection

Michel L. Lapidus	Related
Laurence Raymond Harper, Jr.	Related
Adolfo J. Rumbos	Related
Victor Lenard Shapiro	Related

[222] R. Askey: “What do we do about calculus? First, do no harm,” Am. Math. Monthly 104 : 8 (October 1997), pp. 738–743. In memory of Chih-Han Sah. MR 1543073 article

[223] S. G. Krantz: How to teach mathematics, 2nd edition. American Mathematical Society (Providence, RI), 1998. With an appendix by Richard Askey. book

[224] R. Askey: “Vietoris’s inequalities and hypergeometric series,” pp. 63–76 in Recent progress in inequalities (Niš, Yugoslavia [Serbia], 20–22 June 1996). Edited by G. V. Milovanović. Mathematics and its Applications 430. Kluwer Academic (Dordrecht), 1998. MR 1609992 Zbl 0899.33002 incollection

[225] R. Askey: “A class of non-orthogonal polynomials discovered by F. G. Tricomi,” pp. 123–136 in Tricomi’s ideas and contemporary applied mathematics (Rome, 28–29 November and and Turin, 1–2 December 1997). Atti Convegni Lincei 147. Accademia Nazionale dei Lincei (Rome), 1998. MR 1737493 Zbl 0970.33007 incollection

[226] R. Askey: “Romance in mathematics: The case of S. Ramanujan,” Wisconsin Acad. Rev. 45 : 1 (1998–1999), pp. 13–16. Transcript of a talk given to the Madison Literary Club. article

[227] R. Askey: “The work of George Andrews: A Madison perspective” in The Andrews Festschrift (Maratea, Italy, 31 August–6 September 1998), published as Sém. Lothar. Combin. 42. Issue edited by D. Foata and G.-N. Han. Universität Wien, 1999. Article B42b, 24 pages. MR 1701581 Zbl 1041.01505 incollection

In his own contribution to this volume, George Andrews has touched on a number of themes in his research by looking at the early influences on him of Bailey, Fine, MacMahon, Rademacher and Ramanujan. In this paper, I propose to present a survey of his work organized on a different theme. George has often alluded to the fact that his 1975–76 year in Madison was extremely important in his work. So it seems a reasonable project to survey his career from a Madison perspective. To make this story complete, I begin with Andrews’ work in the late 1960’s that led inexorably to our eventual lengthy collaboration. The year in Madison set in motion three seemingly separate strands of research that were fundamental in much of his subsequent work. Then I briefly describe his collaboration with Rodney Baxter and the discovery of the crank.

George W. Eyre Andrews	Related
Dominique C. Foata	Related
Guo-Niu Han	Related

[228] R. Askey: “Why the Chinese teach math better,” Common Knowl. 12 : 1–2 (1999), pp. 13. article

[229] R. A. Askey, W. Koepf, and T. H. Koornwinder: “Foreword,” pp. 737–738 in Special issue on orthogonal polynomials and computer algebra, published as J. Symb. Comput. 28 : 6. Issue edited by R. A. Askey, W. Koepf, and T. H. Koornwinder. Elsevier (London), December 1999. incollection

Wolfram A. Koepf	Related
Tom Hendrik Koornwinder	Related

[230] R. Askey: “Continued fractions and orthogonal polynomials,” pp. 1–13 in Continued fractions: From analytic number theory to constructive approximation (Columbia, MO, 20–23 May 1998). Edited by B. C. Berndt and F. Gesztesy. Contemporary Mathematics 236. American Mathematical Society (Providence, RI), 1999. MR 1665360 Zbl 0944.33013 incollection

Friedrich Gesztesy	Related
Bruce C. Berndt	Related

[231] G. E. Andrews, R. Askey, and R. Roy: Special functions. Encyclopedia of Mathematics and its Applications 71. Cambridge University Press, 1999. A paperback edition was published in 2000. MR 1688958 Zbl 0920.33001 book

Ranjan Roy	Related
George W. Eyre Andrews	Related

[232] Special issue on orthogonal polynomials and computer algebra, published as J. Symb. Comput. 28 : 6. Issue edited by R. A. Askey, W. Koepf, and T. H. Koornwinder. Elsevier (London), December 1999. MR 1750542 Zbl 0944.00022 book

Wolfram A. Koepf	Related
Tom Hendrik Koornwinder	Related

[233] Special issue dedicated to Richard A. Askey on the occasion of his 65th birthday, I, published as Methods Appl. Anal. 6 : 1. Issue edited by M. E. H. Ismail and D. Stanton. International Press (Somerville, MA), 1999. MR 1801153 Zbl 0928.00069 book

Mourad El Houssieny Ismail	Related
Dennis Warren Stanton	Related

[234] Special issue dedicated to Richard A. Askey on the occasion of his 65th birthday, II, published as Methods Appl. Anal. 6 : 2. Issue edited by M. E. H. Ismail and D. Stanton. International Press (Somerville, MA), 1999. MR 1801154 Zbl 0953.00050 book

Mourad El Houssieny Ismail	Related
Dennis Warren Stanton	Related

[235] Special issue dedicated to Richard A. Askey on the occasion of his 65th birthday, III, published as Methods Appl. Anal. 6 : 3. Issue edited by M. E. H. Ismail and D. Stanton. International Press (Somerville, MA), 1999. MR 1801155 book

Mourad El Houssieny Ismail	Related
Dennis Warren Stanton	Related

[236] G. Gasper, M. E. H. Ismail, T. Koornwinder, P. Nevai, and D. Stanton: “The mathematical contributions of Richard Askey,” pp. 1–18 in \( q \)-Series from a contemporary perspective (South Hadley, MA, 21–25 June 1998). Edited by M. E. H. Ismail and D. W. Stanton. Contemporary Mathematics 254. American Mathematical Society (Providence, RI), 2000. MR 1768919 Zbl 0953.01015 incollection

Paul G. Névai	Related
Dennis Warren Stanton	Related
Mourad El Houssieny Ismail	Related
Tom Hendrik Koornwinder	Related
George Gasper, Jr.	Related

[237] M. E. H. Ismail and D. W. Stanton: “Curriculum vitae of Richard A. Askey,” pp. 19–29 in \( q \)-Series from a contemporary perspective (South Hadley, MA, 21–25 June 1998). Edited by M. E. H. Ismail and D. W. Stanton. Contemporary Mathematics 254. American Mathematical Society (Providence, RI), 2000. MR 1768920 Zbl 0953.01016 incollection

Mourad El Houssieny Ismail	Related
Dennis Warren Stanton	Related

[238] G. E. Andrews, R. Askey, and R. Roy: Special functions, Paperback edition. Encyclopedia of Mathematics and Its Applications 71. Cambridge University Press, 2000. The original hardback edition was published in 1999. Zbl 1075.33500 book

Ranjan Roy	Related
George W. Eyre Andrews	Related

[239] R. Askey: “Good intentions are not enough,” Chapter 8, pp. 163–183 in The great curriculum debate: How should we teach reading and math?. Edited by T. Loveless. Brookings Institution Press (Washington DC), 2001. incollection

[240] R. Askey: “Looking at textbooks,” pp. 106–125 in Knowing and learning mathematics for teaching (Washington DC, 19–21 March 1999). National Academy Press (Washington DC), 2001. incollection

[241] R. Askey, M. E. H. Ismail, and W. Van Assche: “Ted Chihara and his work on orthogonal polynomials,” pp. 1–11 in Proceedings of the fifth international symposium on orthogonal polynomials, special functions and their applications (Patras, Greece, 20–24 September 1999), published as J. Comput. Appl. Math. 133 : 1–2. Issue edited by P. D. Siafarikas. Elsevier (Amsterdam), 2001. MR 1858266 Zbl 0999.01023 incollection

Panayiotis D. Siafarikas	Related
Mourad El Houssieny Ismail	Related
Theodore Seio Chihara	Related
Walter Van Assche	Related

[242] R. Askey: “Ramanujan and hypergeometric and basic hypergeometric series,” pp. 277–324 in Ramanujan: Essays and surveys. Edited by B. C. Berndt and R. A. Rankin. History of Mathematics 22. American Mathematical Society (Providence, RI), 2001. Originally published in Ramanujan international symposium on analysis (1989). MR 1862756 incollection

Robert Alexander Rankin	Related
Srinivasa Iyengar Ramanujan	Related
Bruce C. Berndt	Related

[243] R. A. Askey: “Delving deeper: Fibonacci and Lucas numbers,” Math. Teacher 98 : 9 (January 2005), pp. 610–615. article

[244] R. Askey: “Duality for classical orthogonal polynomials,” J. Comput. Appl. Math. 178 : 1–2 (2005), pp. 37–43. MR 2127868 Zbl 1073.33006 article

[245] R. Askey, M. E. H. Ismail, and E. Koelink: “Mizan Rahman, his mathematics and literary writings,” pp. 1–28 in Theory and applications of special functions (Baltimore, MD, 15–18 January 2003). Edited by M. E. H. Ismail and E. Koelink. Developments in Mathematics 13. Springer (New York), 2005. With contributions by Babu S. Rahman, Raja A. Rahman, Michael Hoare and Steve Milne. MR 2132456 Zbl 1219.01024 incollection

Michael R. Hoare	Related
Mourad El Houssieny Ismail	Related
Hendrik Tjerk (Erik) Koelink	Related
Babu S. Rahman	Related
Raja A. Rahman	Related
Stephen Carl Milne	Related

[246] R. Askey: “Evaluation of Sylvester type determinants using orthogonal polynomials,” pp. 1–16 in Advances in analysis: Proceedings of the 4th international ISAAC congress (Toronto, 11–16 August 2003). Edited by H. G. W. Begehr, R. P. Gilbert, M. E. Muldoon, and M. W. Wong. World Scientific (Hackensack, NJ), 2005. MR 2184920 Zbl 1090.15007 incollection

Heinrich Gottfried Werner Begehr	Related
Martin E. Muldoon	Related
Man Wah Wong	Related
Robert Pertsch Gilbert	Related

[247] R. Askey: “The 1839 paper of permutations: Its relation to the Rodrigues formula and further developments,” Chapter 6, pp. 105–118 in Mathematics and social utopias in France: Olinde Rodrigues and his times. Edited by S. Altmann and E. L. Ortiz. History of Mathematics 28. American Mathematical Society (Providence, RI), 2005. Zbl 1130.01006 incollection

Simon L. Altmann	Related
Olinde Rodrigues	Related
Eduardo L. Ortiz	Related

[248] G. E. Andrews, G. Gasper, and S. Suslov: “Preface,” pp. 5–6 in Askey special issues, published as Ramanujan J. 13 : 1–3. Springer (New York), June 2007. MR 2281153 Zbl 1186.00042 incollection

Sergei K. Suslov	Related
George W. Eyre Andrews	Related
George Gasper, Jr.	Related

[249] R. Askey: “President Lincoln’s mathematics background,” Math. Teacher 104 : 3 (October 2010), pp. 167. article

[250] R. Askey: “The third mathematics education revolution” in Contemporary issues in mathematics education. Edited by E. A. Gavosto, S. G. Krantz, and W. McCallum. MSRI Publications 36. Cambridge University Press, September 2010. incollection

Steven Krantz	Related
Estela Ana Gavosto	Related
William Gordon McCallum	Related

[251] R. Roy, F. W. J. Olver, R. A. Askey, and R. Wong: “Algebraic and analytic methods,” pp. 1–39 in NIST handbook of mathematical functions. Edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. Cambridge University Press (New York), 2010. MR 2655341 incollection

Ronald F. Boisvert	Related
Frank William John Olver	Related
Charles Winthrop Clark	Related
Daniel William Lozier	Related
Roderick Sue-Chuen Wong	Related
Ranjan Roy	Related

[252] R. A. Askey and R. Roy: “Gamma function,” pp. 135–147 in NIST handbook of mathematical functions. Edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. Cambridge University Press (New York), 2010. MR 2655345 incollection

Ronald F. Boisvert	Related
Frank William John Olver	Related
Charles Winthrop Clark	Related
Daniel William Lozier	Related
Ranjan Roy	Related

[253] R. A. Askey and A. B. Olde Daalhuis: “Generalized hypergeometric functions and Meijer \( G \)-function,” pp. 403–418 in NIST handbook of mathematical functions. Edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark. Cambridge University Press (New York), 2010. MR 2655356 incollection

Ronald F. Boisvert	Related
Frank William John Olver	Related
Charles Winthrop Clark	Related
Adrianus Bernardus Olde Daalhuis	Related
Daniel William Lozier	Related

[254] R. Askey: “Completing Brahmagupta’s extension of Ptolemy’s theorem,” pp. 191–197 in The legacy of Alladi Ramakrishnan in the mathematical sciences. Edited by K. Alladi, J. R. Klauder, and C. R. Rao. Springer (New York), 2010. MR 2744262 Zbl 1322.01005 incollection

Calyampudi Radhakrishna Rao	Related
Alladi Ramakrishnan	Related
Krishnaswami Alladi	Related
John Rider Klauder	Related

[255] R. Askey: “Some elementary uses of complex numbers,” Comput. Methods Funct. Theory 10 : 2 (2010), pp. 579–584. MR 2791325 Zbl 1213.00036 article

[256] R. Askey: “Book review: A. Sultan and A. F. Artzt, ‘The mathematics that every secondary school math teacher needs to know’,” Am. Math. Monthly 118 : 8 (2011), pp. 755–762. MR 2856306 article

Alan Sultan	Related
Alice F. Artzt	Related

[257] W. Miller, Jr.: Symmetry and separation of variables, Reprint edition. Encyclopedia of Mathematics and its Applications 4. Cambridge University Press, 2011. With a foreword by Richard Askey. Reprinted edition of 1977 original. Zbl 1223.35006 book

[258] R. Askey, R. Matsuura, and S. Sword: “The inequality of arithmetic and geometric means from multiple perspectives,” J. Math. Teach. Educ. 109 : 4 (November 2015), pp. 314–318. article

Sarah Elizabeth Sword	Related
Ryota Matsuura	Related

[259] Special issue dedicated to Dick Askey on the occasion of his 80th birthday, published as J. Approx. Theory 193. Issue edited by M. Ismail, T. Koornwinder, P. Nevai, W. Van Assche, and O. Warnaar. Elsevier (San Diego, CA), May 2015. book

Paul G. Névai	Related
Mourad El Houssieny Ismail	Related
Tom Hendrik Koornwinder	Related
Sven Ole Warnaar	Related
Walter Van Assche	Related

[260] T. Koornwinder, W. Van Assche, and O. Warnaar: “A tribute to Dick Askey,” pp. 1–3 in Special issue dedicated to Dick Askey on the occasion of his 80th birthday, published as J. Approx. Theory 193. Issue edited by M. Ismail, T. Koornwinder, P. Nevai, W. Van Assche, and O. Warnaar. Elsevier (San Diego, CA), May 2015. MR 3324559 Zbl 1315.01039 ArXiv 1909.10508 incollection

Paul G. Névai	Related
Mourad El Houssieny Ismail	Related
Tom Hendrik Koornwinder	Related
Sven Ole Warnaar	Related
Walter Van Assche	Related

[261] R. Askey: “What content knowledge should we expect in mathematics education?,” pp. 219–231 in Mathematics matters in education: Essays in Honor of Roger E. Howe. Edited by Y. Li, W. J. Lewis, and J. J. Madden. Advances in STEM Education. Springer (Cham, Switzerland), 2018. incollection

Two content topics will be described. One deals with geometric measurement: length, area, and volume. These are important topics and they have not been learned very well. In particular, what types of relations can one consider when dealing with these three aspects of geometric measurement? The second topic is fractions, and different views of what students should be expected to learn. A third topic will be briefly discussed. This is a new book which has many mathematical topics which arose in secondary classes, with comments on the content both as it relates to student learning and sometimes how the mathematics fits into what had previously been studied in earlier grades and also how the ideas fit into later material. The aim of the work was to set up a framework on mathematical content knowledge for teaching mathematics in secondary school. The authors asked for comments from readers, so some will be given.

W. James Lewis	Related
Yeping Li	Related
Roger Evans Howe	Related
James Joseph Madden	Related

[262] R. Askey: “Mourad Ismail,” pp. 1–5 in Frontiers in orthogonal polynomials and \( q \)-series (Orlando, FL, 10–12 May 2015). Edited by M. Z. Nashed and X. Li. Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes 1. World Scientific (Hackensack, NJ), 2018. MR 3791606 Zbl 1414.33002 incollection

Mourad El Houssieny Ismail	Related
Mohammed Zuhair Nashed	Related
Xin Li	Related