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We study, in this paper, a one parameter deformation of the $q-$Laguerreweight function. An investigation is made on the polynomials orthogonal withrespect to such a weight. With the aid of the two compatibility conditionspreviously obtained in \cite{Chen-Ism} and the $q-$analog of a sum ruleobtained in this paper, we derive expressions for the recurrence coefficientsin terms of certain auxiliary quantities, and show that these quantitiessatisfy a pair of first order non linear difference equations. These differenceequations are similar in form to the recognized asymmetric discrete Painlevesystems such as $\alpha q-$P-IV and $\alpha q-$P-V.

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