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The Combinatorics of Laguerre, Charlier, and Hermite Polynomials
Author(s) -
Labelle Jacques,
Yeh Yeong Nan
Publication year - 1989
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm198980125
Subject(s) - laguerre polynomials , mathematics , hermite polynomials , orthogonal polynomials , classical orthogonal polynomials , wilson polynomials , discrete orthogonal polynomials , combinatorial proof , pure mathematics , algebra over a field , hahn polynomials , combinatorics , gegenbauer polynomials
This paper describes several combinatorial models for Laguerre, Charlier, and Hermite polynomials, and uses them to prove combinatorially some classical formulas. The so‐called “Italian limit formula” (from Laguerre to Hermite), the Appel identity for Hermite polynomials, and the two Sheffer identities for Laguerre and Charlier polynomials are proved. We also give bijective proofs of the three‐term recurrences. These three families form the bottom triangle in R. Askey's chart classifying hypergeometric orthogonal polynomials.