On an Integrable System of q-Difference Equations Satisfied by the Universal Characters: Its Lax Formalism and an Application to q-Painlevé Equations

Abstract

The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of q-difference equations satisfied by the universal characters, which is an extension of the q-KP hierarchy and is called the lattice q-UC hierarchy. We describe the lattice q-UC hierarchy as a compatibility condition of its associated linear system (Lax formalism) and explore an application to the q-Painlevé equations via similarity reduction. In particular a higher-order analogue of the q-Painlevé VI equation is presented.

Study of <i>q</i>-Garnier System by Padé Method

“…We note that this kind of the expression (4.19) has already appeared in [28]. By the definition (4.1), it is easy to see the properties for p k as follows: Proof.…”

Two variations on ( A 3 × A 1 × A 1 ) (1) type discrete Painlevé equations

“…-T 1 : q-Drinfeld-Sokolov system q-P (n+1,n+1) [15], -T 2 : Sakai's q-Garnier system [16], -T 3 : Nagao-Yamada's variation of the q-Garnier system [17], -T 4 : Tsuda's q-UC hierarchy [18].…”

Lax pairs of discrete Painlevé equations: (A2+A1)(1)case

“…The search for and construction of Lax pairs of discrete Painlevé equations has been a very active research area and the investigations have been carried out through many approaches. Noteworthy approaches include extensions of Birkhoff's study of linear q-difference equations [7,37,38], periodic-type reductions from ABS equations or the discrete KP/UC hierarchy [5,32,[39][40][41][42][43][44], extensions of Schlesinger transformations [45][46][47], search for linearizable curves in initial-value space [48][49][50], Padé approximation or interpolation [51][52][53] and the theory of orthogonal polynomials [54][55][56][57][58][59].…”