51,541 research outputs found

    q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers

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    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

    On Zudilin's q-question about Schmidt's problem

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    We propose an elemantary approach to Zudilin's q-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a q-analogue of our recent result in [J. Number Theory 132 (2012), 1731--1740] derived from q-Pfaff-Saalschutz identity.Comment: 5 page

    A note on two identities arising from enumeration of convex polyominoes

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    Motivated by some binomial coefficients identities encountered in our approach to the enumeration of convex polyominoes, we prove some more general identities of the same type, one of which turns out to be related to a strange evaluation of 3F2{}_3F_2 of Gessel and Stanton.Comment: 10 pages, to appear in J. Comput. Appl. Math; minor grammatical change

    Some q-analogues of supercongruences of Rodriguez-Villegas

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    We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k} \equiv (-1)^{\frac{p-1}{2}}\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \sum_{k=0}^{p-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\varepsilon)k} &\equiv (-1)^{\frac{p-1}{2}}q^{\frac{(p^2-1)\varepsilon}{4}}\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \varepsilon=\pm1. Several related conjectures are also proposed.Comment: 14 pages, to appear in J. Number Theor
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