Keywords

Sheffer, B-Type, Orthogonal polynomials, Characterizations, Three-term recurrence relation, Generating functions

Abstract

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) + · · · + xk+1Hk(t)] = ∑∞n=0 Pn(x)tn, with Hi(t) = hi,iti + hi,i+1t i+1 + · · · , h1,1 ≠ 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 ≥ 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.

Notes

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Graduation Date

2009

Advisor

Ismail, Mourad

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0002551

URL

http://purl.fcla.edu/fcla/etd/CFE0002551

Language

English

Release Date

May 2009

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

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Included in

Mathematics Commons

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