Current Search: orthogonal polynomials (x)
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Title
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AN INTRODUCTION TO HELLMANN-FEYNMAN THEORY.
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Creator
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Wallace, David, Ismail, Mourad, University of Central Florida
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Abstract / Description
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The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman 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.
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Date Issued
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2005
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Identifier
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CFE0000403, ucf:46349
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Format
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Document (PDF)
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PURL
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http://purl.flvc.org/ucf/fd/CFE0000403
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Title
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THE SHEFFER B-TYPE 1 ORTHOGONAL POLYNOMIAL SEQUENCES.
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Creator
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Galiffa, Daniel, Ismail, Mourad, University of Central Florida
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Abstract / Description
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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
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2009
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Identifier
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CFE0002551, ucf:47655
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Format
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Document (PDF)
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PURL
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http://purl.flvc.org/ucf/fd/CFE0002551
