Some properties of generalized hypergeometric Appell polynomials

On the asymptotics of the rescaled Appell polynomials

We introduce a new representation for the rescaled Appell polynomials and use it to obtain asymptotic expansions to arbitrary order. This representation consists of a finite sum and an integral over a universal contour (i.e. independent of the particular polynomials considered within the Appell family). We illustrate our method by studying the zero attractors for rescaled Appell polynomials. We also discuss the asymptotics to arbitrary order of the rescaled Bernoulli polynomials.

Recursion formulas of q-Appell functions

Recently, Opps, Saad and Srivastava gave the recursion formulas of Appell's function F2. The first author of this paper then established the recursion formulas for Appell functions F1, F2, F3 and F4 by the contiguous relations of hypergeometric series. In this paper, the authors will present the recursion formulas for q-Appell functions $Φ^{(1)}, Φ^{(2)}, Φ^{(3)}$ and $Φ^{(4)}$ as the q-analogies of F1, F2, F3 and F4's relations.

Wronskian Appell Polynomials and Symmetric Functions

We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlevé equations and in the study of exceptional orthogonal polynomials. We determine their derivatives, their average and variance with respect to Plancherel measure, and introduce several recurrence relations. In addition, we prove an integrality conjecture for Wronskian Hermite polynomials previously made by the first and last authors. Our proofs all exploit strong connections with the theory of symmetric functions.

变形二元$q$-Appell多项式

Deformed Bivariate $q$-Appell Polynomials

本文引入变形$q$-Appell型二元多项式集合，研究其代数性质与结构，揭示其与变形齐次多项式的关系，引入相关算子并推导出Mehler和Rogers型公式，最后给出Bernoulli、Euler和Genocchi型多项式实例。

Appell polynomials and their relatives

This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials among polynomials in non-commuting variables. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and "diagram" formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are als…

关于退化Krawtchouk Appell多项式组合性质的研究

On Combinatorial Properties of the degenerate Krawtchouk Appell polynomials

本文研究了退化Krawtchouk Appell多项式的新类别的组合性质，探讨了其与Stirling数、缩放算子、平移算子及正交Bell多项式之间的联系。