Limit transitions will be derived between the five parameter family of Askey-Wilson polynomials, the four parameter family of big q-Jacobi polynomials and the three parameter family of little q-Jacobi polynomials in n variables associated with root system BC. These limit transitions generalize the known hierarchy structure between these families in the one variable case. Furthermore it will be ...

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realize...

In his 1984 proof of the Bieberbach and Milin conjectures de Branges used a positivity result of special functions which follows from an identity about Jacobi polynomial sums that was found by Askey and Gasper in 1973, published in 1976. In 1991 Weinstein presented another proof of the Bieberbach and Milin conjectures, also using a special function system which (by Todorov and Wilf) was realize...

In this paper we give extensions of the mass formula for biweight enumerators and the Jacobi weight enumerators of binary self-dual codes and binary doubly even self-dual codes. For binary doubly even self-dual codes, our formula is expressed in terms of the root system E8 embedded in C4 for biweight enumerators, while the root system D4 is employed for Jacobi weight enumerators. For self-dual ...

Let {Q n (x)}n≥0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product ⟨f, g⟩ = ∫ 1 −1 f(x)g(x)dμα,β(x) + λ ∫ 1 −1 f (x)g(x)dμα+1,β(x) where λ > 0 and dμα,β(x) = (1− x)α(1 + x)βdx with α > −1, β > −1. In this paper we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials {Q n (x)}n. Necessary conditions for ...

Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.

We prove an addition formula for Jacobi functions r ~' ~) (~17_~-~ ) analogous to the known addition formula for Jacobi polynomials. We exploit the positivity of the coefficients in the addition formula by giving the following application. We prove that the product of two Jacobi functions of the same argument has a nonnegative Fourier-Jacobi transform. This implies that the convolution structur...

The Fisher information of the classical orthogonal polynomials with respect to a parameter is introduced, its interest justified and its explicit expression for the Jacobi, Laguerre, Gegenbauer and Grosjean polynomials found.

Let pm(x) = P (λ) m (x)/P (λ) m (1) be the m-th ultraspherical polynomial normalized by pm(1) = 1. We prove the inequality |x|pn(x)−pn−1(x)pn+1(x) ≥ 0, x ∈ [−1, 1], for −1/2 < λ ≤ 1/2. Equality holds only for x = ±1 and, if n is even, for x = 0. Further partial results on an extension of this inequality to normalized Jacobi polynomials are given.

Abstract There are many families of functions on partitions, such as the shifted symmetric functions, for which corresponding q -brackets quasimodular forms. We extend these so that a congruence subgroup. Moreover, we find subspaces -bracket is modular form. These results follow from properties Taylor coefficients strictly meromorphic quasi-Jacobi forms around rational lattice points.