Abstract
S. Ramanujan recorded several modular equations and P role=presentation> P P P - Q role=presentation> Q Q Q theta function identities in his notebooks and lost notebook without recording the proofs. In this paper, we provide an elementary proof of two modular equations and two P role=presentation> P P P - Q role=presentation> Q Q Q theta function identities of level 35, which have been proved by B.C. Berndt using the theory of modular forms.
Highlights
The Gauss series or the ordinary hypergeometric series 2F1 [a, b; c; z] [16,17,18, 21, 26, 27] is defined by ab z a(a + 1)b(b + 1) z22F1 [a, b; c; z] = 1 + c + 1! c(c + 1) !It is well known that the Gauss hypergeometric function 2F1 [a, b; c; z] has many important applications in mathematics, physics, and engineering and many special functions are the particular cases or limiting values of the Gauss hypergeometric function
Berndt proved eighteen of them by employing the theory of theta functions in the spirit of Ramanujan, whereas for the remaining five he used the theory of modular forms
We have considered the positive sign above as XY X1Y1
Summary
The Gauss series or the ordinary hypergeometric series 2F1 [a, b; c; z] [16,17,18, 21, 26, 27] is defined by ab z a(a + 1)b(b + 1) z2.
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