TWO IDENTITIES DERIVABLE FROM THE JACOBI’S TRIPLE-PRODUCT IDENTITY AND THE RAMANUJAN CONTINUED FRACTION

2017 ◽  
Vol 102 (1) ◽  
pp. 243-249
Author(s):  
Mahendra Pal Chaudhary ◽  
Sangeeta Chaudhary ◽  
Junesang Choi
Keyword(s):  
Continued Fraction ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

scholarly journals A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity

2020 ◽  
Vol 108 (122) ◽  
pp. 23-32
Author(s):  
Mahendra Chaudhary
Keyword(s):  
Theta Function ◽  
Open Problem ◽  
Continued Fraction ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Theta Function Identities ◽  
Jacobi's Triple Product Identity

We establish a set of two new relationships involving R?,R? and Rm-functions, which are based on Jacobi?s famous triple-product identity. We, also provide answer for an open problem of Srivastava, Srivastava, Chaudhary and Uddin, which suggest to find an inter-relationships between R?,R? and Rm(m ? N), q-product identities and continued-fraction identities.

scholarly journals Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Liuquan Wang
Keyword(s):  
Analogous Result ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Nonnegative Coefficients ◽  
Jacobi's Triple Product Identity

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they conjectured that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobi's triple product identity in a different manner.

scholarly journals A simple proof of an identity of Ramanujan

1983 ◽  
Vol 34 (1) ◽  
pp. 31-35 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
Simple Proof ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Jacobi's Triple Product Identity

AbstractOne of Ramanujan's unpublished, unproven identities has excited considerable interest over the years. Indeed, no fewer than four proofs have appeared in the literature. The object of this note is to present yet another proof, simpler than the others, relying only on Jacobi's triple product identity.

scholarly journals A simple proof of Jacobi's four-square theorem

1982 ◽  
Vol 32 (1) ◽  
pp. 61-67 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Triple Product ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Four Square ◽  
Jacobi's Triple Product Identity

AbstractA celebrated result, due to Jacobi, says that the number of representations of the positive integer n as a sum of four squares is equal to eight times the sum of the divisors of n which are not divisible by 4. We give a new and simple proof of this result which depends only on Jacobi's triple product identity.

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