# A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials

@article{Kim2001ACF,
title={A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials},
author={Dongsu Kim and Jiang Zeng},
journal={Eur. J. Comb.},
year={2001},
volume={22},
pages={313-332}
}
• Published 2001
• Mathematics, Computer Science
• Eur. J. Comb.
We prove a formula for the linearization coefficients of the general Sheffer polynomials, which unifies all the special known results for Hermite, Charlier, Laguerre, Meixner and Meixner?Pollaczek polynomials. Furthermore, we give a new and explicit real version of the corresponding formula for Meixner?Pollaczek polynomials. Our proof is based on some explicit bijections and sign-reversing weight-preserving involutions.

#### Figures and Topics from this paper

Nonnegative linearization coefficients of the generalized Bessel polynomials
In this work, we solve the general linearization problem for the generalized Bessel polynomials using their inversion formula. For some particular values, we get a recurrence relation satisfied byExpand
Combinatorics of generalized Tchebycheff polynomials
• Mathematics, Computer Science
• Eur. J. Comb.
• 2003
By considering a family of orthogonal polynomials generalizing the Tchebycheff polynomials of the second kind we refine the corresponding results of De Sainte-Catherine and Viennot on TchebycheffExpand
Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 2013
This work proposes a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials using the method of separation of variables and applies it to determine the number of perfect matchings, derangements, and other weighted permutation problems. Expand
Linearization coefficients for Sheffer polynomial sets via lowering operators
• Mathematics, Computer Science
• Int. J. Math. Math. Sci.
• 2006
This paper expresses explicitly the linearization coefficients for polynomial sets of Sheffer type using the corresponding lowering operators. Expand
Linearization coefficients for orthogonal polynomials using stochastic processes
Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients haveExpand
Linearization coefficients for Boas-Buck polynomial sets
• Mathematics, Computer Science
• Appl. Math. Comput.
• 2007
The linearization coefficients related to three Boas–Buck polynomial sets using their corresponding generating functions are expressed explicitly and some well-known results as particular cases as well as some new reduction formulae are obtained. Expand
Product formulas on posets, Wick products, and a correction for the $q$-Poisson process
We give an example showing that the product and linearization formulas for the Wick product versions of the $q$-Charlier polynomials in (Anshelevich 2004) are incorrect. Next, we observe that theExpand
The combinatorics of Al-Salam-Chihara q-Laguerre polynomials
• Mathematics, Computer Science
• 2011
Combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients of the Al-Salam-Chihara q-Laguerre polynomial are described. Expand
Appell polynomials and their relatives
This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can beExpand
Asymptotics of generalized derangements
• Mathematics, Computer Science
• 2013
We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics ofExpand

#### References

SHOWING 1-10 OF 37 REFERENCES
Weighted Derangements and the Linearization Coefficients of Orthogonal Sheffer Polynomials
The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite,Expand
The Combinatorics of q-Hermite polynomials and the Askey - Wilson Integral
• Computer Science, Mathematics
• Eur. J. Comb.
• 1987
The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of AskeyExpand
Combinatorial interpretation of integrals of products of hermite, Laguerre and Tchebycheff polynomials
• Mathematics
• 1985
Certain integrals of products of Laguerre polynomials have been interpreted as numbers of generalized derangements by Kaplansky, Even, Gillis, Jackson, Askey, Ismail, and Rashed. The analog for theExpand
The Combinatorics of Meixner Polynomials: Linearization Coefficients
Various aspects of the Meixner polynomials are described, including combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients. Expand
The Combinatorics of q-Charlier Polynomials
• Mathematics, Computer Science
• J. Comb. Theory, Ser. A
• 1995
Abstract We describe various aspects of the Al Salam-Carlitz q -Charlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearizationExpand
Generalized Rook Polynomials and Orthogonal Polynomials
We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the sameExpand
Laguerre Polynomials, Weighted Derangements, and Positivity
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 1988
A calculation of the linearization coefficients of the Laguerre polynomials is proposed by means of analytic and combinatorial methods to extend to the case of an arbitrary $\alpha$ a combinatoric and analytic result due to Askey, Ismail, and Koornwinder and Even and Gillis. Expand
Enumerations of permutations and continued J -fractions
Abstract In this paper, we evaluate the generating function of the symmetric group with respect to five statistics. The continued fraction expansion of its ordinary generating function is thenExpand
Permutation problems and special functions
• Mathematics
• 1976
Abstract : Integral representations for some combinatorial numbers are established and a combinatorial interpretation is found for certain sums involving the Meixner polynomials. Various integrals ofExpand
A note on tensor products of q -algebra representations and orthogonal polynomials
• Mathematics
• 1996
Abstract We work out examples of tensor products of distinct generalized slq(2) algebras with a factor from the positive discrete series of representations of one algebra and a factor from theExpand