Overpartitions related to the mock theta function $\omega (q)$

George E. Andrews¹ ; Atul Dixit² ; Daniel Schultz¹ ; Ae Ja Yee¹

¹ Department of Mathematics The Pennsylvania State University University Park, PA 16802, U.S.A.
² Department of Mathematics Indian Institute of Technology Gandhinagar Palaj, Gandhinagar, Gujarat 382355, India

Acta Arithmetica, Tome 181 (2017) no. 3, pp. 253-286

Résumé

It was recently shown that $q\omega (q)$, where $\omega (q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega }(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_{\omega }(n)$, and express its generating function in terms of a ${}_3\phi _{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven-parameter $q$-series identity which generalizes a deep identity due to the first author as well as its generalization by R. P. Agarwal. We also derive two interesting congruences satisfied by the overpartition analogue, and some congruences satisfied by the associated smallest parts function.

Détail

DOI : 10.4064/aa161225-7-11

Keywords: recently shown omega where omega third order mock theta functions generating function nbsp omega number partitions positive integer odd parts twice smallest part paper study overpartition analogue nbsp omega express its generating function terms phi basic hypergeometric series infinite series involving little q jacobi polynomials accomplished obtaining seven parameter q series identity which generalizes deep identity due first author its generalization nbsp nbsp agarwal derive interesting congruences satisfied overpartition analogue congruences satisfied associated smallest parts function