The Sheffer B-Type 1 Orthogonal Polynomial Sequences

Download or read book The Sheffer B-Type 1 Orthogonal Polynomial Sequences written by Daniel Joseph Galiffa and published by . This book was released on 2009 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one type. Sheffer extensively developed properties of the B-Type 0 polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary B-Type k by constructing the generalized generating function A(t)exp[xH1(t)+ ... + x[superscript k +1]H[subscript k](t)] = [summation sign] [infinity sign above summation sign] [subscript n=0 below summation sign]P[subscript n](x)t[superscript n], with H[subscript i](t) = h[subscript i], [subscript i]t[superscript i] + h[subscript i], [subscript i+1]t[superscript i+1] + ..., h1, 1 [does not equal] 0. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher (k [greater than or equal to] 1). We present a preliminary analysis of a special case of the B-Type 1 (k = 1) class, which is an extension of the B-Type 0 class, in order to determine which sets, if any, are also orthogonal sets. Lastly, we consider an extension of this research and comment on future considerations. In this work the utilization of computer algebra packages is indispensable, as computational difficulties arise in the B-Type 1 class that are unlike those in the B-Type 0 class.