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 Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Matthieu Josuat-Vergès ◽  
Jang-Soo Kim
Keyword(s):  
Functional Equation ◽  
Triple Product ◽  
Combinatorial Proof ◽  
Jacobi’S Triple Product Identity ◽  
Product Identity ◽  
Nous Montrons ◽  
Schröder Paths ◽  
International Audience ◽  
Finite Sum ◽  
Jacobi's Triple Product Identity

International audience We give a combinatorial proof of a Touchard-Riordan-like formula discovered by the first author. As a consequence we find a connection between his formula and Jacobi's triple product identity. We then give a combinatorial analog of Jacobi's triple product identity by showing that a finite sum can be interpreted as a generating function of weighted Schröder paths, so that the triple product identity is recovered by taking the limit. This can be stated in terms of some continued fractions called T-fractions, whose important property is the fact that they satisfy some functional equation. We show that this result permits to explain and generalize some Touchard-Riordan-like formulas appearing in enumerative problems. Nous donnons une preuve combinatoire d'une formule à la Touchard-Riordan due au premier auteur. En conséquence, nous faisons appara\^ıtre un lien entre cette formule et l'identité du produit triple de Jacobi. Nous donnons un analogue combinatoire à l'identité du produit triple en montrant qu'une somme finie peut être interprétée comme fonction génératrice de chemins de Schröder pondérés, de sorte que l'identité du produit triple s'obtient en passant à la limite. Ceci peut être énoncé en termes de fractions continues appelées T-fractions, dont la propriété importante est le fait qu'elle satisfont certaines équations fonctionnelles. Nous montrons que ce résultat permet d'expliquer et généraliser certaines formules à la Touchard-Riordan apparaissant dans des problèmes d'énumération.

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.

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