Showing papers by "Richard Askey published in 1974"
TL;DR: In this article, a new proof is given for two inequalities for trigonometric polynomials, and some applications and new proof are given for some applications of these inequalities for polynomial numbers.
Abstract: : Some applications and a new proof are given for two inequalities for trigonometric polynomials. (Author)
74 citations
TL;DR: For ultraspherical series, this article showed that their power series has nonnegative coefficients for ρ = 1 − 1 − ρ + ρ − 1 \leqq x \LEqq 1.
Abstract: The functions $(1 - r)^{ - 2|\lambda |} (1 - 2xr + r^2 )^{ - \lambda } $ are shown to be absolutely monotonic, or equivalently, that their power series have nonnegative coefficients for $ - 1 \leqq x \leqq 1$. One consequence is a simple proof of Kogbetliantz’s theorem on positive Cesaro summability for ultraspherical series, [7].
36 citations
TL;DR: The first of a series of papers which will give simple proofs of a number of recent formulas for Jacobi polynomials was given in this article, where one of Gegenbauer's proofs for his integral representation of ultraspherical polynomial is given, and then a fractional integration gives Koornwinder's integral representation.
Abstract: This is the first of a series of papers which will give simple proofs of a number of recent formulas for Jacobi polynomials. In this paper one of Gegenbauer’s proofs for his integral representation of ultraspherical polynomials is given, and then a fractional integration gives Koornwinder’s integral representation for Jacobi polynomials. This is then combined with Koornwinder’s product formula to give a new proof of a bilinear sum of Bateman.
24 citations
TL;DR: In this paper, it was shown that a Jacobi polynomial with nonnegative coefficients has nonnegative power series coefficients for α = 0, 1, 2, \cdots $.
Abstract: Dunkl’s recent expression of a certain Jacobi polynomial times a simple polynomial as a sum of ultraspherical polynomials with nonnegative coefficients is translated into a result between Jacobi polynomials of the same argument and then applied to prove that \[\frac{1}{{\{ (1 - r)(1 - s)(1 - t)[(1 - r)(1 - s)(1 + t) + (1 - r)(1 + s)(1 - t) + (1 + r)(1 - s)(1 - t)]\} ^{(\alpha + 1)/2 }}}\] has nonnegative power series coefficients for $\alpha = 0,1,2, \cdots $.
15 citations
01 Jan 1974
TL;DR: For a series with positive Cesaro means of order -y, this paper showed that its Abel means have a positive CSA for 0 s r < (a + 1)/(y + 1), -1 < a < Y.
Abstract: If a series has positive Cesaro means of order -y, then its Abel means have positive Cesaro means of order a for 0 s r < (a + 1)/(y + 1), -1 < a
2 citations