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ISRN Discrete Mathematics
Volume 2011 (2011), Article ID 476462, 16 pages
http://dx.doi.org/10.5402/2011/476462
Research Article

Differential Equation and Recursive Formulas of Sheffer Polynomial Sequences

Department of Mathematics, University of St. Thomas, 2115 Summit Avenue, Saint Paul, MN 55105-1079, USA

Received 3 August 2011; Accepted 7 September 2011

Academic Editors: M. Chlebík, K. Eriksson, and M. C. Wilson

Copyright © 2011 Heekyung Youn and Yongzhi Yang. 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 derive a differential equation and recursive formulas of Sheffer polynomial sequences utilizing matrix algebra. These formulas provide the defining characteristics of, and the means to compute, the Sheffer polynomial sequences. The tools we use are well-known Pascal functional and Wronskian matrices. The properties and the relationship between the two matrices simplify the complexity of the generating functions of Sheffer polynomial sequences. This work extends He and Ricci's work (2002) to a broader class of polynomial sequences, from Appell to Sheffer, using a different method. The work is self-contained.