scholarly journals q-Hypergeometric Proofs of Polynomial Analogues of the Triple Product Identity, Lebesgue?s Identity and Euler?s Pentagonal Number Theorem

2005 ◽  
Vol 8 (4) ◽  
pp. 467-474 ◽  
Author(s):  
S. Ole Warnaar
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Pentagonal Number Theorem

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.

Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Chuanan Wei ◽  
Dianxuan Gong
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Pentagonal Number Theorem ◽  
Jacobi Triple Product Identity ◽  
Liouville’S Theorem

AbstractBy means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity.

scholarly journals Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

10.37236/1796 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
James A. Sellers ◽  
Andrew V. Sills ◽  
Gary L. Mullen
Keyword(s):  
Famous Theorem ◽  
Partition Identities ◽  
Product Identity ◽  
Powers Of 2 ◽  
Pentagonal Number Theorem ◽  
Series Product

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

scholarly journals Consequences of a sextuple-product identity

1987 ◽  
Vol 10 (3) ◽  
pp. 545-549
Author(s):  
John A. Ewell
Keyword(s):  
Triple Product ◽  
Product Identity ◽  
Special Cases ◽  
Jacobi Triple Product Identity

A sextuple-product identity, which essentially results from squaring the classical Gauss-Jacobi triple-product identity, is used to derive two trigonometrical identities. Several special cases of these identities are then presented and discussed.

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