A generalization of the Hirschhorn-Farkas-Kra septagonal numbers identity (Q5937453)

This paper is about some well-known combinatorial identities, more specific a generalization of the quintuple product identity discovered by Farkas and Kra and independently by Hirschhorn is proven. The proof of the generalization starts from the Jacobi triple product identity and uses only elementary arithmetic.NEWLINENEWLINENEWLINEThe paper is a bit sloppy: on page 114 the author mentions that taking \(k=3\) in Theorem 1 corresponds to the Farkas-Kra-Hirschhorn identity, but this must be \(k=5\). Furthermore on page 116 the \((-1)^{n+ j+r}\) in the formulas involving three summation signs has to be split into \((-1)^n\) and \((-1)^{j+ r}\), and the factor \((-1)^n\) has to be taken into the third summation sign (it can not be pulled out to the first two summation signs). The last formula on page 116 obviously lacks a factor \(q^{j(kj+ 2r+ 2-k)/2}\).