Table of Contents
Journal of Discrete Mathematics
Volume 2013 (2013), Article ID 450481, 10 pages
http://dx.doi.org/10.1155/2013/450481
Research Article

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

Department of Mathematics, Uppsala University, P.O. Box 480, 751 06 Uppsala, Sweden

Received 19 June 2012; Accepted 2 December 2012

Academic Editor: Franck Petit

Copyright © 2013 Thomas Ernst. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 .