Geodesic
The arithmetic nature of the triple and quintuple product identities
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 140-148.

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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. 
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N. V. Budarina; V. A. Bykovskii. The arithmetic nature of the triple and quintuple product identities. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 140-148. http://geodesic.mathdoc.fr/item/DVMG_2011_11_2_a1/

[1] B. C. Berndt, Ramanujan's notebooks, v. III, Springer-Verlag, New York, 1991, 510 pp. | MR | Zbl

[2] D. Foata, Guo-Niu Han, “The triple, quintuple and septuple product identities revisited”, Sém. Lothar. Combin., 42 (1999), Art. B42o, 12 pp. (electronic) | MR

[3] E. V. Wright, “An enumerative proof of an identity of Jacobi”, J. London Math. Soc., 40 (1965), 55–57 | DOI | MR | Zbl

[4] C. Jr. Sudler, “Two enumerative proofs of an identity of Jacobi”, Proc. Edinburgh Math. Soc. (2), 15 (1966), 67–71 | DOI | MR | Zbl

[5] George E. Andrews, “A simple proof of Jacobi's triple product identity”, Proc. Amer. Math. Soc., 16 (1965), 333–334 | DOI | MR

[6] B. Gordon, “Some identities in combinatorial analysis”, Quart. J. Math. Oxford, Ser. (2), 12 (1961), 285–290 | DOI | MR | Zbl

[7] I. G. Macdonald, “Affine root systems and Dedekind's $\eta$-function”, Invent. Math., 15 (1972), 91–143 | DOI | MR | Zbl

[8] J. Soviet. Math., 41:2 (1988), 889–891 | DOI | MR | Zbl | Zbl

[9] Herbert S. Wilf, “The number-theoretic content of the Jacobi triple product identity”, The Andrews Festschrift (Maratea, 1998), Sém. Lothar. Combin., 42, 1999, Art. B42k, 4 pp. (electronic) | MR

[10] B. A. Venkov, Elementarnaya teoriya chisel, ONTI NKPT SSSR, 1937, 222 pp.

[11] J. V. Uspensky, M. A. Heaslet, Elementary Number Theory, McGraw-Hill Book Company, Inc., New York, 1939, 484 pp. | MR

[12] Kenneth S. Williams, Number theory in the spirit of Liouville, London Mathematical Society Student Texts, 76, Cambridge University Press, Cambridge, 2011, 287 pp. | MR | Zbl