‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers |
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Authors: | Ana F Loureiro Jiang Zeng |
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Institution: | 1. School of Mathematics, Statistics & Actuarial Science (SMSAS), Cornwallis Building, University of Kent, Canterbury, U.K.;2. +33 (0)472431984+33 (0)472431984;3. Université de Lyon, Université Claude Bernard Lyon 4. 1, Institut Camille Jordan, Villeurbanne cedex, France |
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Abstract: | 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. |
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Keywords: | Orthogonal polynomials q‐classical polynomials q‐differential equations q‐Jacobi– Stirling numbers q‐Stirling numbers signed partitions 33C45 05A10 (34B24 34L05) |
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