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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 $$S_2=\{n\mid n\equiv \pm 1,\pm 4, 6 \pmod{12}\}$$.

##### MSC:
 11P81 Elementary theory of partitions 05A19 Combinatorial identities, bijective combinatorics
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##### References:
 [1] R. Blecksmith, The Determination of Ramanujan Pairs, Ph.D. Dissertation, University of Arizona, 1983. [2] Richard Blecksmith, John Brillhart, and Irving Gerst, A computer-assisted investigation of Ramanujan pairs, Math. Comp. 46 (1986), no. 174, 731 – 749. · Zbl 0586.10008 [3] Richard Blecksmith, John Brillhart, and Irving Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), no. 177, 29 – 38. · Zbl 0617.10010 [4] Richard Blecksmith, John Brillhart, and Irving Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301 – 314. · Zbl 0767.11048 [5] L. Carlitz and M. V. Subbarao, A simple proof of the quintuple product identity, Proc. Amer. Math. Soc. 32 (1972), 42 – 44. · Zbl 0234.05005 [6] John A. Ewell, Completion of a Gaussian derivation, Proc. Amer. Math. Soc. 84 (1982), no. 2, 311 – 314. · Zbl 0484.05010 [7] John A. Ewell, Some combinatorial identities and arithmetical applications, Rocky Mountain J. Math. 15 (1985), no. 2, 365 – 370. Number theory (Winnipeg, Man., 1983). · Zbl 0585.10034 [8] G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, Oxford, 1965. · Zbl 0020.29201
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