Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations

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...

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...

Soliton Solutions for Quasilinear Schrödinger Equations

Finding All Hypergeometric Solutions of Linear Diierential Equations Finding All Hypergeometric Solutions of Linear Diierential Equations Finding All Hypergeometric Solutions of Linear Diierential Equations

Hypergeometric sequences are such that the quotient of two successive terms is a xed rational function of the index. We give a generalization of M. Petkov sek's algorithm to nd all hypergeometric sequence solutions of linear recurrences, and we describe a program to nd all hypergeometric functions that solve a linear diierential equation. Solutions hyperg eom etriques des equations dii erentiel...