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

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 . 1. Introduction In this paper we will introduce the -Pascal and -Wronskian matrices in a general setting, with the aim of fruitful applications for -Appell polynomials. These two matrices contain a certain -difference operator, just like the -deformed Leibniz functional matrix from [1]. By the -Leibniz formula, products of such matrices contain a certain operator , just as in the previous article, but with a slightly different definition. However, since in almost all equations we compute the function value at , this operator will convert to ordinary matrix multiplication by formula (6). There is a connection to the -Pascal matrix [2] for the special case -exponential function. 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. Carlitz cleverly observed that the generating function for Appell polynomials can be inverted when . The NWA -addition seldom occurs in the literature, except for the author’s papers; it was first seen in an Italian paper by Nalli [4]. Corresponding to the two dual -additions, NWA and JHC, there are two dual -Bernoulli polynomials, with the same names; these are special cases of -Appell polynomials, the content of Section 2. It turns out that for certain purposes we will need pseudo -Appell polynomials, which correspond to the other basic -addition, JHC; these polynomials were first seen in [5]. The -Appell numbers form a certain algebraic structure, with two operations, which was first described by Al-Salam [6]. This structure will in a certain way be generalized to -Appell polynomials. The -Bernoulli