A study of the associated linear problem forq-PV
Christopher M. Ormerod · Journal of Physics A Mathematical and Theoretical · 2010
We consider the associated linear problem for a q -analogue of the fifth Painlevé equation ( q -P V ). We identify a lattice of connection preserving deformations in the space of the characteristic data for the linear problem with the lattice of translational Bäcklund transformations for q -P V ; hence, show all translational Bäcklund transformations possess a Lax pair. We show that the big q -Laguerre polynomials, and a suitable generalization, solve a special case of the linear problem, and hence, find solutions to q -P V in terms of determinants of Hankel matrices with entries consisting of rational or hypergeometric functions. We show that we may specialize the connection preserving deformations to rational deformations of the weight and the recurrence relations.