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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     author = {N. V. Budarina and V. A. Bykovskii},
     title = {The arithmetic nature of the triple and quintuple product identities},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
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     number = {2},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2011_11_2_a1/}
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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/