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 $\alpha q-$P-IV and $\alpha q-$P-V. 
Y.Chen and J. Griffin
math.CA