Non linear difference equations arising from a deformation of the q-Laguerre weight

We study, in this paper, a one parameter deformation of the $q-$Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions previously obtained in \cite{Chen-Ism} and the $q-$analog of a sum rule obtained in this paper, we derive expressions for the recurrence coefficients in terms of certain auxiliary quantities, and show that these quantities satisfy a pair of first order non linear difference equations. These difference equations are similar in form to the recognized asymmetric discrete Painleve systems such as $αq-$P-IV and $αq-$P-V.

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