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Some infinite product identities. (English) Zbl 0655.10010
The authors derive power series expansions of four infinite products of the form \(\displaystyle \prod_{n\in S_ 1}(1-x^ n)\prod_{n\in S_ 2}(1+x^ n),\) where the index sets \(S_ 1\) and \(S_ 2\) are specified with respect to a modulus. A useful formula for expanding the products of two Jacobi triple products is established and nonexistence results for identities of the two forms are also proved. A typical result is the identity \[ \prod^{\infty}_{\substack{n=1 \\ n\in S_1}} (1-x^n)\prod^{\infty}_{\substack{n=1 \\ n\in S_2}} (1+x^n)=\sum^{\infty}_{n=0} (x^{2n(n+1)}+x^{6n(n+1)+1}); \]
here \(S_1=\{n\mid n\equiv 0,\pm 7,\pm 5\pmod{24}\}\) and \(S_2=\{n\mid n\equiv \pm 1,\pm 4, 6 \pmod{12}\}\).

11P81 Elementary theory of partitions
05A19 Combinatorial identities, bijective combinatorics
Full Text: DOI
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