25,516 research outputs found

    Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

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    Let pΛ‰(n)\bar{p}(n) denote the number of overpartitions of nn. It was conjectured by Hirschhorn and Sellers that \bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40) for nβ‰₯0n\geq 0. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for pΛ‰(40n+35)\bar{p}(40n+35) modulo 5. Using the (p,k)(p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence \bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5). Combining this congruence and the congruence \bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8) obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.Comment: 11 page

    The q-WZ Method for Infinite Series

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    Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the 6Ο•5_6\phi_5 sum, Ramanujan's 1ψ1_1\psi_1 sum and Bailey's 6ψ6_6\psi_6 sum.Comment: 17 page

    Interlacing Log-concavity of the Boros-Moll Polynomials

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    We introduce the notion of interlacing log-concavity of a polynomial sequence {Pm(x)}mβ‰₯0\{P_m(x)\}_{m\geq 0}, where Pm(x)P_m(x) is a polynomial of degree m with positive coefficients ai(m)a_{i}(m). This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of Pm(x)P_m(x) interlace the ratios of consecutive coefficients of Pm+1(x)P_{m+1}(x) for any mβ‰₯0m\geq 0. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.Comment: 10 page
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