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Nonlinear difference equations for the generalized little q-laguerre polynomials
Chen H.1; Filipuk G.2; Chen Y.1
2017
Source PublicationJournal of Difference Equations and Applications
ISSN15635120 10236198
Volume23Issue:12Pages:1943-1973
Abstract

In this paper, a study is made on polynomials orthogonal with respect to the generalized little q-Laguerre weight, defined by w(x)=x(qx;q)∞, 0< x <1. Here and (a;q):=∏ (1−aq). This weight is supported on the exponential lattice {1,q,q,…,q,…}. Let the subleading coefficient of x of the monic polynomials be δ. From the q-analogue of the ladder operators, the associated supplementary conditions and a ‘sum rule’, we deduce a system of difference equations satisfied by δ. This system is used to obtain the first few terms in a formal asymptotic expansion of δ. We express the recurrence coefficients in terms of this subleading coefficient and show that the first few terms in the formal expansions in powers of q agree with the first few terms for the corresponding expansions of the recurrence coefficients in the classical case. Moreover, we find certain non-linear difference equations for the recurrence coefficients of the monic polynomials, auxiliary functions in the ladder operators and for δ. We also observe the phenomenon of singularity confinement, related to that observed in theq-discrete Painlevé equations. Furthermore, we give a generalization of the weight function, characterized by w(x/q)/w(x)=Ax+Bx+C for A≠0,C≠1A≠0,C≠1 on the exponential lattice. In this situation we find another system of difference equations satisfied by δ and study its behaviour for large n. The paper ends with the discussion on a deformation of the generalized little q-Laguerre weight.

KeywordAsymptotic Expansions Difference Equations Orthogonal Polynomials Singularity Confinement
DOI10.1080/10236198.2017.1380004
URLView the original
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000423267500003
PublisherTAYLOR & FRANCIS LTD, 2-4 PARK SQUARE, MILTON PARK, ABINGDON OR14 4RN, OXON, ENGLAND
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Citation statistics
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Affiliation1.Universidade de Macau
2.Uniwersytet Warszawski
Recommended Citation
GB/T 7714
Chen H.,Filipuk G.,Chen Y.. Nonlinear difference equations for the generalized little q-laguerre polynomials[J]. Journal of Difference Equations and Applications,2017,23(12):1943-1973.
APA Chen H.,Filipuk G.,&Chen Y..(2017).Nonlinear difference equations for the generalized little q-laguerre polynomials.Journal of Difference Equations and Applications,23(12),1943-1973.
MLA Chen H.,et al."Nonlinear difference equations for the generalized little q-laguerre polynomials".Journal of Difference Equations and Applications 23.12(2017):1943-1973.
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