On the recurrence coefficients for the q-Laguerre weight and discrete Painlevé equations

Abstract We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the q -Laguerre weight on the degree parameter n . We show that this dependence is described by a discrete Painlevé equation on the family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> Sakai surfaces, but this equation is different from the standard examples of discrete Painlevé equations of this type and instead is a composition of two such. This case study is a good illustration of the effectiveness of a recently proposed geometric identification scheme for discrete Painlevé equations.