A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials

Kim, Dong–Su; Zeng, Jiang

doi:10.1006/eujc.2000.0459

Cited by 18 publications

(11 citation statements)

References 15 publications

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“…A typical question in this direction is to find explicitly the linearization coefficients R P n 1 (x)P n 2 (x) · · · P n k (x)dµ(x). (1.4) Already the proofs of the positivity of these coefficients are quite subtle [19] and they are known explicitly only in very rare cases [28].…”

Section: Introductionmentioning

confidence: 99%

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Anshelevich¹

2004

Internat. Math. Res. Notices

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Downloaded from (both sides considered as formal power series). Many classical (nonorthogonal) polynomial families are Appell. They arise in finite operator calculus [40] and the study of hypergroups, in numerical analysis (Bernoulli polynomials are Appell), and also in probability theory, in the study of stochastic processes [32], noncentral limit theorems [13, 25], and natural exponential families [39]. From the combinatorial point of view, they have nice linearization and multinomial formulas.The third family of polynomials has not apparently been explicitly defined before, although it appears implicitly in the paper [43]. For this reason, we will call them the Kailath-Segall polynomials. These are polynomials in (infinitely many) variables A whole theory of free probability [49], based on this notion, is by now quite well developed. It turns out that there are "free analogs of" the Appell and Kailath-Segall polynomials, which, very roughly, are obtained by replacing commuting variables with

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Section: Introductionmentioning

confidence: 99%

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Anshelevich¹

2004

Internat. Math. Res. Notices

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“…When the β-extension of MacMahon's Master theorem is combined with the exponential formula [30,33], all the known combinatorial interpretations of the linearization coefficients of the orthogonal Sheffer polynomials can be deduced by computing their generating functions. Another way to gain insight into the combinatorial interpretation of the linearization coefficients is from their corresponding moment sequences, see [24,32,35], (1.8) µ n = R x n dµ(x).…”

Section: Introductionmentioning

confidence: 99%

Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

Ismail

Kasraoui

Zeng

2013

Journal of Combinatorial Theory, Series A

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We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of separation of variables. We illustrate our approach by applying it to determine the number of perfect matchings, derangements, and other weighted permutation problems. The separation of variables technique naturally leads to integral representations of combinatorial numbers where the integrand contains a product of one or more types of orthogonal polynomials. This also establishes the positivity of such integrals.2010 Mathematics Subject Classification. Primary 33D15, 05A15, Secondary 30E05, 33 C15.

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“…Several q-analogues of the recurrence relation (2) and moments (3) were investigated in the last two decades (see [2,18,19]) in order to obtain new mahonian statistics on the symmetric groups. On the other hand, in view of the unified combinatorial interpretations of several aspects of Sheffer orthogonal polynomials (moments, polynomials, and the linearization coefficients)(see [14,20,22]) it is natural to seek for a q-version of this picture.…”

Section: Introductionmentioning

confidence: 99%

The combinatorics of Al-Salam–Chihara q-Laguerre polynomials

Kasraoui¹,

Stanton²,

Zeng³

2011

Advances in Applied Mathematics

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We describe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Grassmann cells.

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A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials

Cited by 18 publications

References 15 publications

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Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

The combinatorics of Al-Salam–Chihara q-Laguerre polynomials

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