Proof of a Conjecture of Hirschhorn and Sellers on Overpartitions

Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that \bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40) for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-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

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 $_6\phi_5$ sum, Ramanujan's $_1\psi_1$ sum and Bailey's $_6\psi_6$ sum.Comment: 17 page

Interlacing Log-concavity of the Boros-Moll Polynomials

We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. 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