Representations of $q$-orthogonal polynomials

Abstract

The linearization problem is the problem of finding the coefficients $C_k(m,n)$ in the expansion of the product $P_n\left(x\right)Q_m\left(x\right)$ of two polynomial systems in terms of a third sequence of polynomials $R_k\left(x\right)$,

$$P_n\left(x\right)Q_m\left(x\right)=\sum _{k=0}^{n+m}{C}_k(m,n)R_k\left(x\right)\text{.}$$

Note that, in this setting, the polynomials $P_n,Q_m$ and $R_k$ may belong to three different polynomial families. If $Q_m\left(x\right)=1$, we are faced with the so-called connection problem, which for $P_n\left(x\right)=x^n$ is known as the inversion problem for the family $R_k\left(x\right)$.

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

$$\sigma \left(x\right)D_q{D}_{1/q}{P}_n\left(x\right)+\tau \left(x\right)D_q{P}_n\left(x\right)+{\lambda }_n{P}_n\left(x\right)=0\text{,}$$

where the $q$-differential operator $D_q$ is defined by

$$D_{q}f\left(x\right)=\frac{f\left(qx\right)-f\left(x\right)}{(q-1)x}\text{.}$$