-Pascal and -Wronskian Matrices with Implications to -Appell Polynomials

Thomas Ernst

doi:10.1155/2013/450481

Abstract

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

Highlights

In this paper we will introduce the q-Pascal and q-Wronskian matrices in a general setting, with the aim of fruitful applications for q-Appell polynomials
Since in almost all equations we compute the function value at t = 0, this operator will convert to ordinary matrix multiplication by formula (6)
To motivate the generalized q-Pascal functional matrix, we show that the Carlitz [3, page 247] formula for the product of generating functions for Appell polynomials has a q-analogue involving the NWA q-addition

Summary

Introduction

In this paper we will introduce the q-Pascal and q-Wronskian matrices in a general setting, with the aim of fruitful applications for q-Appell polynomials These two matrices contain a certain q-difference operator, just like the q-deformed Leibniz functional matrix from [1]. The Appell polynomials are seldom encountered in the literature; one exception was Carlitz’ article [3], where a formula for the product of two generating functions was given. Appell showed that multiplication of two Appell polynomials is commutative He considered the inverse of An as a natural generalization of ordinary division. To motivate the generalized q-Pascal functional matrix, we show that the Carlitz [3, page 247] formula for the product of generating functions for Appell polynomials has a q-analogue involving the NWA q-addition. By the umbral definition of q-Appell polynomials we have k∑=n0(kn)qΦ󸀠n−k,qΦk,q (x). Formulas (32) and (34) follow by the commutativity and the associativity of NWA

Specializations

Conclusion