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q ‐differential equations for q ‐classical polynomials and q ‐Jacobi–Stirling numbers
Author(s) -
Loureiro Ana F.,
Zeng Jiang
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400381
Subject(s) - stirling numbers of the first kind , stirling number , bell polynomials , mathematics , stirling numbers of the second kind , differential operator , eigenfunction , reciprocal , operator (biology) , jacobi polynomials , pure mathematics , algebra over a field , set (abstract data type) , orthogonal polynomials , computer science , physics , linguistics , chemistry , repressor , quantum mechanics , transcription factor , gene , programming language , eigenvalues and eigenvectors , philosophy , biochemistry
We introduce, characterise and provide a combinatorial interpretation for the so‐called q ‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q ‐differential operator having the q ‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q ‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q ‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.