Representations of q-orthogonal polynomials

The linearization problem is the problem of finding the coefficients C"k(m,n) in the expansion of the product P"n(x)Q"m(x) of two polynomial systems in terms of a third sequence of polynomials R"k(x), P"n(x)Q"m(x)=@?k=0n+mC"k(m,n)R"k(x). Note that, in this setting, the polynomials P"n,Q"m and R"k may belong to three different polynomial families. If Q"m(x)=1, we are faced with the so-called connection problem, which for P"n(x)=x^n is known as the inversion problem for the family R"k(x). In this paper, we use an algorithmic approach to compute the connection and linearization coefficients between orthogonal polynomials of the q-Hahn tableau. These polynomial systems are solutions of a q-differential equation of the type @s(x)D"qD"1"/"qP"n(x)+@t(x)D"qP"n(x)+@l"nP"n(x)=0, where the q-differential operator D"q is defined by D"qf(x)=f(qx)-f(x)(q-1)x.

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