×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.