Octabasic Laguerre polynomials and permutation statistics
type: Item
instance of: scholarly article
title: Octabasic Laguerre polynomials and permutation statistics
zbMATH Open document ID: 0877.05062
DOI: 10.1016/0377-0427(95)00250-2
author: Rodica Simion
author: D. Stanton
published in: Journal of Computational and Applied Mathematics
publication date: 1997-12-08T00:00:00Z
review text: A set of orthogonal polynomials is studied which depends on ``eight independent \(q\)'s.'' If all ``eight \(q\)'s'' are set equal to 1, these orthogonal polynomials specialize to Laguerre polynomials. Using Viennot's combinatorial theory of orthogonal polynomials, it is shown that the moments for these polynomials can be expressed combinatorially, in several different ways, as certain generating functions for permutations. This opens the path for a study of these polynomials in terms of statistics for various combinatorial objects. It is this what the remainder of the paper is devoted to. Specializations of the octabasic orthogonal polynomials and their combinatorics are systematically studied. These specializations include numerous classical families of orthogonal polynomials (besides Laguerre polynomials) and related \(q\)-analogues. The combinatorics involves permutations, of course, set partitions, noncrossing partitions, involutions, etc., and various statistics on these objects. Particularly attractive and useful are the many links that are worked out, again in a systematic manner, between the various combinatorial objects.
Mathematics Subject Classification ID: 05E35
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 33D70
Mathematics Subject Classification ID: 05A18
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 33C65
zbMATH DE Number: 908397
zbMATH Keywords: Charlier polynomials
zbMATH Keywords: Chebychev polynomials
zbMATH Keywords: Hermité polynomials
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: generating functions
zbMATH Keywords: permutations
zbMATH Keywords: statistics
zbMATH Keywords: \(q\)-analogues
zbMATH Keywords: set partitions
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00250-2
OpenAlex ID: W2063962974
cites work: Congruences for the q-secant numbers
cites work: Divisibility Properties of the q-Tangent Numbers
cites work: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
cites work: Associated Laguerre and Hermite polynomials
cites work: Permutations depending on the exceedance type and the number of inversions and combinatorial interpretation of a continued fraction of Heine
cites work: Some \(q\)-analogues of the Schröder numbers arising from combinatorial statistics on lattice paths
cites work: On Co-Recursive Orthogonal Polynomials
cites work: Q4170993
cites work: The combinatorics of \(q\)-Charlier polynomials
cites work: Q5820721
cites work: Combinatorial aspects of continued fractions
cites work: Non-overlapping partitions, continued fractions, Bessel functions and a divergent series
cites work: Q3344211
cites work: q-Catalan numbers
cites work: Q4040884
cites work: The q-Stirling numbers of first and second kinds
cites work: Q3669422
cites work: The zeros of basic Bessel functions, the functions J(nu+ax)(x), and associated orthogonal polynomials
cites work: Linear birth and death models and associated Laguerre and Meixner polynomials
cites work: On the Generalized Chebyshev Polynomials
cites work: On Constructing Distribution Functions; with Applications to Lommel Polynomials and Bessel Functions
cites work: Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers
cites work: The q-analogue of the Laguerre polynomials
cites work: The Distribution of Crossings of Chords Joining Pairs of 2n Points on a Circle
cites work: Combinatorial statistics on non-crossing partitions
cites work: On \(q\)-analogues of partially ordered sets
cites work: Specializations of Generalized Laguerre Polynomials
cites work: Q3715122
cites work: \(p,q\)-Stirling numbers and set partition statistics
description: scientific article; zbMATH DE number 908397