Pairs of partitions without repeated odd parts (Q439277)

The authors prove the identities NEWLINE\[NEWLINE \sum_{n \geq 0} \frac{(x,1/x)_nq^n}{(zq,q/z)_n} = \frac{(1-z)^2}{(1-z/x)(1-xz)} + \frac{z(x,1/x)_{\infty}}{(1-z/x)(1-xz)(zq,q/z)_{\infty}} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \sum_{n \geq 0} \frac{(x,q/x)_nq^n}{(z,q/z)_{n+1}} = \frac{1}{x(1-z/x)(1-q/(xz))} + \frac{(x,q/x)_{\infty}}{z(1-x/z)(1-q/(xz))(z,q/z)_{\infty}}. NEWLINE\]NEWLINE Here we have used standard \(q\)-series notation. The proofs depend on previous work of the first author [Proc. Lond. Math. Soc. (3) 91, No. 3, 598--622 (2005; Zbl 1089.33012)] together with the Watson-Whipple transformation. A number of corollaries are recorded, some of which contain the infinite product NEWLINE\[NEWLINE \frac{(-q;q^2)_{\infty}^2}{(zq^2,q^2/z;q^2)_{\infty}^2}. NEWLINE\]NEWLINE This leads to a discussion of ranks for partition pairs without repeated odd parts. The authors define three of these, describe what each counts, and show that each provides a combinatorial interpretation for the congruence NEWLINE\[NEWLINE tt(3n+2) \equiv 0 \pmod{3}, NEWLINE\]NEWLINE where \(tt(n)\) is the number of partition pairs of \(n\) without repeated odd parts.