### Far Eastern Mathematical Journal

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The arithmetic nature of the triple and quintuple product identities

N. V. Budarina, V. A. Bykovskii

2011, issue 2, P. 140–148

Abstract
In this paper the new proof is suggested for decomposition of twisted with quadratic characters modulo 4 and 3 theta-functions to the infinite product. It is based on the Euler's method of logarithmic derivation and the elementary arithmetic concepts.

Keywords:
theta-function, Liouville identities, infinite product

#### References

[1] B. C. Berndt, Ramanujan's notebooks, v. III, Springer-Verlag, New York, 1991, 510 pp.
[2] D. Foata, Guo-Niu Han, “The triple, quintuple and septuple product identities revisited”, Se?m. Lothar. Combin., 42 (1999), Art. B42o, 12 pp. (electronic).
[3] E. V. Wright, “An enumerative proof of an identity of Jacobi”, J. London Math. Soc., 40 (1965), 55–57.
[4] C. Jr. Sudler, “Two enumerative proofs of an identity of Jacobi”, Proc. Edinburgh Math. Soc. (2), 15 (1966), 67–71.
[5] George E. Andrews, “A simple proof of Jacobi's triple product identity”, Proc. Amer. Math. Soc., 16 (1965), 333–334.
[6] B. Gordon, “Some identities in combinatorial analysis”, Quart. J. Math. Oxford, Ser. (2), 12 (1961), 285–290.
[7] I. G. Macdonald, “Affine root systems and Dedekind's $\eta$-function”, Invent. Math., 15 (1972), 91–143.
[8] A. M. Vershik, “A bijective proof of the Jacobi identity, and reshapings of the Young diagrams”, Differentsialnaya Geometriya, Gruppy Li i Mekh. VIII, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 155, 1986, 3–6, 193; translation in J. Soviet. Math., 41:2 (1988), 889–891.
[9] Herbert S. Wilf, “The number-theoretic content of the Jacobi triple product identity”, The Andrews Festschrift (Maratea, 1998), Se?m. Lothar. Combin., 42, 1999, Art. B42k, 4 pp. (electronic).
[10] B. A. Venkov, Elementarnaia teoriia chisel, ONTI NKPT SSSR, 1937, 222 s.
[11] J. V. Uspensky, M. A. Heaslet, Elementary Number Theory, McGraw-Hill Book Company, Inc., New York, 1939, 484 pp.
[12] Kenneth S. Williams, Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge University Press, Cambridge, 2011, 287 pp.

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