scholarly journals Two Finite Forms of Watson's Quintuple Product Identity and Matrix Inversion

10.37236/1078 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
X. Ma
Keyword(s):  
Matrix Inversion ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Special Cases

Recently, Chen-Chu-Gu and Guo-Zeng found independently that Watson's quintuple product identity follows surprisingly from two basic algebraic identities, called finite forms of Watson's quintuple product identity. The present paper shows that both identities are equivalent to two special cases of the $q$-Chu-Vandermonde formula by using the ($f,g$)-inversion.

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.

scholarly journals A simple proof of the quintuple product identity

1972 ◽  
Vol 32 (1) ◽  
pp. 42-42 ◽  
Author(s):  
L. Carlitz ◽  
M. V. Subbarao
Keyword(s):  
Simple Proof ◽  
Quintuple Product Identity ◽  
Product Identity

scholarly journals A generalisation of the quintuple product identity

1988 ◽  
Vol 44 (1) ◽  
pp. 42-45 ◽  
Author(s):  
M. D. Hirschhorn
Keyword(s):  
General Identity ◽  
Quintuple Product Identity ◽  
Product Identity

AbstractThe quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

scholarly journals Ramanujan's remarkable summation formula as a $2$-papameter generalization of the quintuple product identity

2002 ◽  
Vol 33 (3) ◽  
pp. 285-288
Author(s):  
S. Bhargava ◽  
Chandrashekar Adiga ◽  
M. S. Mahadeva Naika
Keyword(s):  
Summation Formula ◽  
Triple Product ◽  
Binomial Theorem ◽  
Suitable Transformation ◽  
Quintuple Product Identity ◽  
Product Identity

It is well known that `Ramanujan's remarkable summation formula' unifies and generalizes the $q$-binomial theorem and the triple product identity and has numerous applications. In this note we will demonstrate how, after a suitable transformation of the series side, it can be looked upon as a $2$-parameter generalization of the quintuple product identity also.

scholarly journals A new form of the quintuple product identity and its application

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1869-1873
Author(s):  
Bhaskar Srivastava
Keyword(s):  
Simple Proof ◽  
Continued Fractions ◽  
Direct Application ◽  
General Identity ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
New Form

We give a new form of the quintuple product identity. As a direct application of this new form a simple proof of known identities of Ramanujan and also new identities for other well known continued fractions are given. We also give and prove a general identity for (q3m; q3m)?.

Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type

10.37236/1190 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Peter Paule
Keyword(s):  
Short Proof ◽  
Similar Type ◽  
Quintuple Product Identity ◽  
Product Identity ◽  
Computer Proofs ◽  
Creative Telescoping ◽  
Ramanujan Identity

New short and easy computer proofs of finite versions of the Rogers-Ramanujan identities and of similar type are given. These include a very short proof of the first Rogers-Ramanujan identity that was missed by computers, and a new proof of the well-known quintuple product identity by creative telescoping.

scholarly journals Another simple proof of the quintuple product identity

2005 ◽  
Vol 2005 (15) ◽  
pp. 2511-2515 ◽  
Author(s):  
Hei-Chi Chan
Keyword(s):  
Simple Proof ◽  
Triple Product ◽  
Quintuple Product Identity ◽  
Product Identity

We give a simple proof of the well-known quintuple product identity. The strategy of our proof is similar to a proof of Jacobi (ascribed to him by Enneper) for the triple product identity.

scholarly journals The quintuple product identity and shifted partition functions

1996 ◽  
Vol 68 (1-2) ◽  
pp. 3-13 ◽  
Author(s):  
Krishnaswami Alladi
Keyword(s):  
Partition Functions ◽  
Quintuple Product Identity ◽  
Product Identity

ON THE SCHRÖTER FORMULA FOR THETA FUNCTIONS

2009 ◽  
Vol 05 (08) ◽  
pp. 1477-1488 ◽  
Author(s):  
ZHI-GUO LIU ◽  
XIAO-MEI YANG
Keyword(s):  
Theta Function ◽  
Theta Functions ◽  
Modular Equations ◽  
Trigonometric Identity ◽  
Product Identity ◽  
Function Identity ◽  
Special Cases ◽  
Theta Function Identity ◽  
Addition Formulas

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.

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