19 citations · 25 across the 14 of their papers we have counts for
14 papers
New proofs of theorems on q-orthogonal functions
Dandan Chen, Zhiguo Liu
In this paper, we establish a -integral formula by using the orthogonality relation, and also provide a new proof of the -orthogonality relation for the continuous -ultras…
On a Transformation of Triple -Series and Rogers-Hecke Type Series
Zhi-Guo Liu
Using the method of the -exponential differential operator, we give an extension of the Sears transformation formula. Based on this extended formula and a -series exp…
A universal identity for theta functions of degree eight and applications
Zhi-Guo Liu
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition form…
Kronecker theta function and a decomposition theorem for theta functions I
Zhi-Guo Liu
The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's summation. Using the Kronecker theta function as build…
Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions
Zhi-Guo Liu, Nian Hong Zhou
In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular,…
Askey--Wilson polynomials and a double -series transformation formula with twelve parameters
Zhi-Guo Liu
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integr…
A -extension of a partial differential equation and the Hahn polynomials
Zhi-Guo Liu
Using the theory of analytic functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of -partial differential equat…
Gauss summation and Ramanujan type series for
Zhi-Guo Liu
Using some properties of the gamma function and the well-known Gauss summation formula for the classical hypergeometric series, we prove a four-parameter series expansion formula,…
Extensions of Ramanujan's reciprocity theorem and the Andrews--Askey integral
Zhi-Guo Liu
Ramanujan's reciprocity theorem may be considered as a three-variable extension of Jacobi's triple product identity. Using the method of -partial differential equations, we exte…
A -summation and the orthogonality relations for the -Hahn polynomials and the big -Jacobi polynomials
Zhi-Guo Liu
Using a general -summation formula, we derive a generating function for the -Hahn polynomials, which is used to give a complete proof of the orthogonality relation for the $q…
On the -derivative and -series expansions
Zhi-Guo Liu
Using a general -series expansion, we derive some nontrivial -formulas involving many infinite products. A multitude of Hecke--type series identities are derived. Some genera…
On the -partial differential equations and -series
Zhi-Guo Liu
Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of -partial differential equations, the…
On a system of -partial differential equations with applications to -series
Zhi-Guo Liu
Using the theory of functions of several variables and -calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of -pa…
On the complex Hermite polynomials and partial differential equations
Zhi-Guo Liu
In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a s…