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…