A symmetric sequence of orthogonal polynomials associated with the Stieltjes-Wigert polynomials (Q861088)

The authors consider the polynomial set \(\left\{V_n(x;q,r)\right\}_{n\geq 0}\) defined by \[ V_n(x;q,r)=\sum_{k=0}^n(-1)^k\left[\begin{matrix} n\cr k\end{matrix}\right]_q q^{k(k+r)}x^k \] where \(r\) is a free real parameter and \(q\in(0,1)\). It contains as special cases the Stieltjes-Wigert polynomials and the Rogers-Szegö polynomials. They show that the polynomials \(V_n(x;e^{{1\over 2b}},a+{1\over 2})\) in which \(b<0\) and \(a\in \mathbb R\) are orthogonal with respect to the weight function \(x^{(a+b\ln x)}\) on \([0,\infty)\). Then, by means of the polynomials \(V_n(x;q,r)\), they define the symmetric polynomial set \(\{U_n(x;q,r)\}_{n\geq 0}\) by \[ U_{2n}(x;q,r)=V_n(x^2;q,r)\quad \text{and } \quad U_{2n+1}(x;q,r)=xV_n(x^2;q,r). \] They show that the polynomials \(U_n(x;e^{{2\over \beta}},{\alpha\over 2})\) in which \(\beta<0\) and \(\alpha\in \mathbb R\) are orthogonal with respect to the weight function \(| x| ^{(\alpha+\beta\ln | x| }\) on \((-\infty,\infty)\).