Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

Mourad E.H. Ismail; Anisse Kasraoui; Jiang Zeng

doi:10.1016/j.jcta.2012.10.007

Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

Mourad E.H. Ismail
, Anisse Kasraoui
, Jiang Zeng

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

14 Citations (Scopus)

Abstract

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. © 2012 Elsevier Inc.

Original language	English
Pages (from-to)	561-599
Journal	Journal of Combinatorial Theory - Series A
Volume	120
Issue number	3
DOIs	https://doi.org/10.1016/j.jcta.2012.10.007
Publication status	Published - Apr 2013
Externally published	Yes

Bibliographical note

Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

Funding

1 Research supported by NPST Program of King Saud University; project number 10-MAT1293-02 and King Saud University in Riyadh and by Research Grants Council of Hong Kong under contract #101410. 2 Research supported by the grant S9607-N13 from Austrian Science Foundation FWF in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. 3 Research supported by the grant ANR-08-BLAN-0243-03.

Research Keywords

Derangements
Linearization coefficients
Orthogonal polynomials
Q-Analogues
Separation of variables
Sheffer polynomials

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title = "Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials",

abstract = "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. {\textcopyright} 2012 Elsevier Inc.",

keywords = "Derangements, Linearization coefficients, Orthogonal polynomials, Q-Analogues, Separation of variables, Sheffer polynomials",

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note = "Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].",

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month = apr,

doi = "10.1016/j.jcta.2012.10.007",

language = "English",

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pages = "561--599",

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AU - Ismail, Mourad E.H.

AU - Kasraoui, Anisse

AU - Zeng, Jiang

N1 - Publication details (e.g. title, author(s), publication statuses and dates) are captured on an “AS IS” and “AS AVAILABLE” basis at the time of record harvesting from the data source. Suggestions for further amendments or supplementary information can be sent to [email protected].

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N2 - 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. © 2012 Elsevier Inc.

AB - 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. © 2012 Elsevier Inc.

KW - Derangements

KW - Linearization coefficients

KW - Orthogonal polynomials

KW - Q-Analogues

KW - Separation of variables

KW - Sheffer polynomials

UR - https://www.scopus.com/pages/publications/84869447377

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U2 - 10.1016/j.jcta.2012.10.007

DO - 10.1016/j.jcta.2012.10.007

M3 - RGC 21 - Publication in refereed journal

SN - 0097-3165

VL - 120

SP - 561

EP - 599

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 3

ER -

Separation of variables and combinatorics of linearization coefficients of orthogonal polynomials

Abstract

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Research Keywords

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