A new semi-finite form of the quintuple product identity
The quintuple product identity are deduced from a new semi-finite form, which are obtained from the very-well-poised $_6ϕ_5$ series.
math.CO↗
arXiv subjects
Jun-Ming Zhu
Publications and source records attributed to Jun-Ming Zhu.
Generalizations of a terminating summation formula of basic hypergeometric series and their applications
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula. And, an identity unifying Jacobi's triple product identity and the quintuple product identity is obtained as a special case of the bilateral one.
math.CO↗
A semi-finite proof of Jacobi's triple product identity
Jacobi's triple product identity is proved from one of Euler's $q$-exponential functions in an elementary way.
math.HO↗
An alternate circular summation formula of theta functions and its applications
We prove a general alternate circular summation formula of theta functions, which implies a great deal of theta-function identities. In particular, we recover several identities in Ramanujan's Notebook from this identity. We also obtain two formulaes for $(q;q)_\infty^{2n}$.
math.CO↗
A note on a generalized circular summation formula of theta functions
In this note, we make a correction of the imaginary transformation formula of Chan and Liu's circular formula of theta functions. We also get the imaginary transformation formulaes for a type of generalized cubic theta functions.
math.CO↗
The number of convex pentagons and hexagons in an $n$-triangular net
In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.
math.CO↗
The solutions of four $q$-functional equations
In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.
math.CO↗