k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

Hypergeometric series solutions of linear operator equations

Elliptic Hypergeometric Solutions to Elliptic Difference Equations⋆

It is shown how to define difference equations on particular lattices {xn}, n ∈ Z, made of values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations have remarkable simple interpolatory expansions. Only linear difference equations of first order are considered here.

Hypergeometric Solutions to Ultradiscrete Painlevé Equations

We propose new solutions to ultradiscrete Painlevé equations that cannot be derived using the ultradiscretization method. In particular, we show the third ultradiscrete Painelevé equation possesses hypergeometric solutions. We show this by considering a lift of these equations to a non-archimedean valuation field in which we may relax the subtraction free framework of previous explorations of t...

Solution of Some Integral Equations Involving Confluent k-Hypergeometric Functions

The principle aim of this research article is to investigate the properties of k-fractional integration introduced and defined by Mubeen and Habibullah [1], and secondly to solve the integral equation of the form             1 1 0 , ; d , ; k x k k x t g x F t x f t k                t t 0, 0, 0,0 k x , for          , where     1 1 , ; , ; k F x k     ...

Hypergeometric solutions to the q-Painlevé equations

It is well-known that the continuous Painlevé equations PJ (J=II,. . .,VI) admit particular solutions expressible in terms of various hypergeometric functions. The coalescence cascade of hypergeometric functions, from the Gauss hypergeometric function to the Airy function, corresponds to that of Painlevé equations, from PVI to PII [1]. The similar situation is expected for the discrete Painlevé...