Journal of Integer Sequences, Vol. 24 (2021), Article 21.2.1

Combinatorial Properties of a Generalized Class of Laguerre Polynomials

Mark Shattuck
Department of Mathematics
University of Tennessee
Knoxville, TN 37996


In this paper, we consider various combinatorial aspects of a family of polynomials, denoted by Ln(α,β) (x), whose coefficients Sα,β(n,k) correspond to a special case of the partial r-Bell polynomials. Among the particular cases of Ln(α,β)(x) are the generalized Laguerre polynomials, associated Lah polynomials, and polynomials arising in the study of hyperbolic partial differential equations. Here we provide a combinatorial treatment of Ln(α,β)(x) and its coefficients, which were studied previously strictly from an algebraic standpoint. In addition to providing combinatorial proofs of some prior identities, we derive several new relations using the combinatorial interpretations for Ln(α,β) (x) and Sα,β(n,k). Our proofs make frequent use of sign-changing involutions on various weighted structures. Finally, we introduce a bivariate polynomial generalization arising as a distribution for a pair of statistics and establish some of its basic properties.

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(Concerned with sequences A000110 A000262 A008275 A008277 A008297.)

Received November 2 2020; revised version received January 16 2021. Published in Journal of Integer Sequences, January 23 2021.

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