Extending theorems of Göllnitz, a new family of partition identities
Let \(A_{k,i}(n)\) denote the number of partitions of \(n\) into parts that are \(\equiv 5-2i\pmod 4\) and are \(\geq 4k+5-2i\), or are \(\equiv 10-4i\pmod 8\) and are \(< 8k+ 10- 4i\), or are \(\equiv 4i-2\pmod 8\). Let \(B_{k,i}(n)\) denote the number of partitions of \(n\) in distinct parts where the odd ones are \(\equiv 5-2i\pmod 4\) and are \(\geq 4k+5-2i\). Let \(C_{k,i}(n)\) denote the number of partitions of \(n\) in distinct parts, \(n= \alpha_1+\cdots+ \alpha_t\), where if \(\alpha_j\) is odd then \(\alpha_j\geq 4k+5-2i\), and \(\alpha_j- \alpha_{j+\ell}\neq 2\), \(4k+3-2i\), \(4k+1-2i\). Then it is shown that, if \(i=1\) or 2 and \(k\geq 1\), \(A_{k,i}(n)= B_{k,i}(n)= C_{k,i}(n)\). These results generalise results of \textit{H. Göllnitz} and \textit{B. Gordon} [see \textit{G. E. Andrews}, The theory of partitions, Addison-Wesley (1976; Zbl 0655.10001), Section 7.4].
scientific article; zbMATH DE number 1406507