Nonlinear difference equations for the generalized little q-Laguerre polynomials

In this paper, a study is made on polynomials orthogonal with respect to the generalized little q-Laguerre weight, defined by w(x)=xα(q2x2;q2)∞, 0<x<1. Here 0<q<1, α>0 and (a;q)∞:=∏∞k=0(1−aqk). This weight is supported on the exponential lattice {1,q,q2,…,qk,…}. Let the subleading coefficient of xn−1 of the monic polynomials be δn. From the q-analogue of the ladder operators, the associated supplementary conditions and a ‘sum rule’, we deduce a system of difference equations satisfied by δn. This system is used to obtain the first few terms in a formal asymptotic expansion of δn. We express the recurrence coefficients in terms of this subleading coefficient and show that the first few terms in the formal expansions in powers of qn agree with the first few terms for the corresponding expansions of the recurrence coefficients in the classical case. Moreover, we find certain non-linear difference equations for the recurrence coefficients of the monic polynomials, auxiliary functions in the ladder operators and for δn. We also observe the phenomenon of singularity confinement, related to that observed in the q-discrete Painlevé equations. Furthermore, we give a generalization of the weight function, characterized by w(x/q)/w(x)=Ax2+Bx+C for A≠0,C≠1 on the exponential lattice. In this situation we find another system of difference equations satisfied by δn and study its behaviour for large n. The paper ends with the discussion on a deformation of the generalized little q-Laguerre weight.