Current Search: orthogonal polynomials (x)


Title

AN INTRODUCTION TO HELLMANNFEYNMAN THEORY.

Creator

Wallace, David, Ismail, Mourad, University of Central Florida

Abstract / Description

The HellmannFeynman theorem is presented together with certain allied theorems. The origin of the HellmannFeynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.

Date Issued

2005

Identifier

CFE0000403, ucf:46349

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0000403


Title

THE SHEFFER BTYPE 1 ORTHOGONAL POLYNOMIAL SEQUENCES.

Creator

Galiffa, Daniel, Ismail, Mourad, University of Central Florida

Abstract / Description

In 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one $type$. Sheffer extensively developed properties of the $B$\emph{Type 0} polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary $B$\emph{Type k} by constructing the generalized generating function $A(t)\mathrm=\sum_^\infty$, with $H_i(t)=h_t^i+h_t^+\cdots,\phantomh_\neq 0$. Although extensive research has been...
Show moreIn 1939, I.M. Sheffer proved that every polynomial sequence belongs to one and only one $type$. Sheffer extensively developed properties of the $B$\emph{Type 0} polynomial sequences and determined which sets are also orthogonal. He subsequently generalized his classification method to the case of arbitrary $B$\emph{Type k} by constructing the generalized generating function $A(t)\mathrm=\sum_^\infty$, with $H_i(t)=h_t^i+h_t^+\cdots,\phantomh_\neq 0$. Although extensive research has been done on characterizing polynomial sequences, no analysis has yet been completed on sets of type one or higher ($k\geq1$). We present a preliminary analysis of a special case of the $B$\emph{Type 1} ($k=1$) class, which is an extension of the $B$\emph{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 $B$\emph{Type 1} class that are unlike those in the $B$\emph{Type 0} class.
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Date Issued

2009

Identifier

CFE0002551, ucf:47655

Format

Document (PDF)

PURL

http://purl.flvc.org/ucf/fd/CFE0002551