Celebratio Mathematica

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.

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

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.

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

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

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