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

Effect of pyramiding Bt and CpTI genes on resistance of cotton to Helicoverpa armigera (Lepidoptera: Noctuidae) under laboratory and field conditions

Transgenic cotton (Gossypium hirsutum L.) varieties, adapted to China, have been bred that express two genes for resistance to insects. the Cry1Ac gene from Bacillus thuringiensis (Berliner) (Bt), and a trypsin inhibitor gene from cowpea (CpTI). Effectiveness of the double gene modification in conferring resistance to cotton bollworm, Helicoverpa armigera (Hübner) (Lepidoptera: Noctuidae), was studied in laboratory and field experiments. In each experiment, performance of Bt+CpTI cotton was compared with Bt cotton and to a conventional nontransgenic variety. Larval survival was lower on both types of transgenic variety, compared with the conventional cotton. Survival of first-, second-, and third-stage larvae was lower on Bt+CpTI cotton than on Bt cotton. Plant structures differed in level of resistance, and these differences were similar on Bt and Bt+CpTI cotton. Likewise, seasonal trends in level of resistance in different plant structures were similar in Bt and Bt+CpTI cotton. Both types of transgenic cotton interfered with development of sixth-stage larvae to adults, and no offspring was produced by H. armigera that fed on Bt or Bt+CpTI cotton from the sixth stage onward. First-, second-, and third-stage larvae spent significantly less time feeding on transgenic cotton than on conventional cotton, and the reduction in feeding time was significantly greater on Bt+CpTI cotton than on Bt cotton. Food conversion efficiency was lower on transgenic varieties than on conventional cotton, but there was no significant difference between Bt and Bt+CpTI cotton. In 3-yr field experimentation, bollworm densities were greatly suppressed on transgenic as compared with conventional cotton, but no significant differences between Bt and Bt+CpTI cotton were found. Overall, the results from laboratory work indicate that introduction of the CpTI gene in Bt cotton raises some components of resistance in cotton against H. armigera, but enhanced control of H. armigera under field conditions, due to expression of the CpTI gene, was not demonstrate