An Application of Jacobi Triple Product Identity in Integer Partition Theory

2022 ◽  
Vol 12 (01) ◽  
pp. 41-46
Author(s):  
万里 马
Keyword(s):  
Triple Product ◽  
Integer Partition ◽  
Partition Theory ◽  
Product Identity ◽  
Jacobi Triple Product Identity

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.

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 QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA

2011 ◽  
Vol 04 (01) ◽  
pp. 31-34
Author(s):  
S. Bhargava ◽  
Chandrashekar Adiga ◽  
M. S. Mahadeva Naika
Keyword(s):  
Summation Formula ◽  
Triple Product ◽  
Binomial Theorem ◽  
Interesting Fact ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Special Case

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

2009 ◽  
Vol 05 (05) ◽  
pp. 765-778 ◽  
Author(s):  
SHAUN COOPER
Keyword(s):  
Eisenstein Series ◽  
Congruence Subgroup ◽  
Gauss Sums ◽  
Triple Product ◽  
Simple Construction ◽  
Dedekind Eta Function ◽  
Modular Transformation ◽  
Product Identity ◽  
Jacobi Triple Product Identity ◽  
Extra Difficulty

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

Cubic Analogues of the Jacobian Theta Function θ(z, q)

1993 ◽  
Vol 45 (4) ◽  
pp. 673-694 ◽  
Author(s):  
Michael Hirschhorn ◽  
Frank Garvan ◽  
Jon Borwein
Keyword(s):  
Hypergeometric Function ◽  
Theta Function ◽  
Modular Forms ◽  
Elliptic Function ◽  
Triple Product ◽  
Product Identity ◽  
Jacobi Triple Product Identity

AbstractThere are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity

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