Celebratio Mathematica

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

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

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

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

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

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Richard Allen Askey

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