Representations of qq-orthogonal polynomials

The linearization problem   is the problem of finding the coefficients Ck(m,n)Ck(m,n) in the expansion of the product Pn(x)Qm(x)Pn(x)Qm(x) of two polynomial systems in terms of a third sequence of polynomials Rk(x)Rk(x), Pn(x)Qm(x)=∑k=0n+mCk(m,n)Rk(x). Note that, in this setting, the polynomials Pn,Qm and RkRk may belong to three different polynomial families. If Qm(x)=1Qm(x)=1, we are faced with the so-called connection problem  , which for Pn(x)=xnPn(x)=xn is known as the inversion problem   for the family Rk(x)Rk(x).In this paper, we use an algorithmic approach to compute the connection and linearization coefficients between orthogonal polynomials of the qq-Hahn tableau. These polynomial systems are solutions of a qq-differential equation of the type σ(x)DqD1/qPn(x)+τ(x)DqPn(x)+λnPn(x)=0,σ(x)DqD1/qPn(x)+τ(x)DqPn(x)+λnPn(x)=0, where the qq-differential operator DqDq is defined by Dqf(x)=f(qx)−f(x)(q−1)x.