On the rational solutions of q-Painlevé V equation

Abstract

Abstract. We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.

Special Polynomials Associated with the Noumi-Yamada System of Type A5(1)

“…It is natural to ask what kind of solutions to the Noumi-Yamada system of type A ð1Þ 5 one can get by starting with the fixed points with respect to the transformation p 3 . In this setting, we find that the system is reduced to…”

“…In fact, it has been revealed that the universal characters appear associated with a class of algebraic solutions to P VI and the Garnier systems [4,11]. It is also known that the rational solutions to q-P V are expressed in terms of a q-analogue of the universal characters [3].…”

New connection formulae for someq-orthogonal polynomials inq-Askey scheme

“…The continuous q-Laguerre polynomials had manifested their apparition in the rational solutions of the q-analogue of Painlevé V differential equation [9], namely as the entries of its associated determinant. They are defined by: [5] …”

Universal Characters and q-Painlevé Systems

“…We often denote the q-Painlevé system (0.1) by q-P (A N −1 ). Note that q-P (A 2 ) and q-P (A 3 ) coincide with the q-Painlevé equations q-P IV and q-P V respectively; see [1,7].…”

“…As a consequence of this remarkable fact, (0.1) admits a class of rational solutions in terms of Schur polynomials; see [3] or Theorem 5.1 below. On the other hand, for the case N = 4 (q-P V ), Masuda [7] discovered another class of rational solutions which contains the former one in terms of the universal characters. Here the universal character is a generalization of Schur polynomial attached to a pair of partitions; see [5].…”