Orthogonality relations for Al-Salam-Carlitz polynomials of type II (Q2343165)

Using a special case of Askey's \(q\)-beta integral evaluation formula, the author determines the orthogonality relations for the Al-Salam-Carlitz polynomials of type II with respect to a family of measures supported on a discrete subset \(\mathbb R_q = \mathbb R_q(z_-,z_+) = z_-q^{\mathbb Z} \cup z_+ q^{\mathbb Z}\) of \(\mathbb R\) where \(z_-<0\) and \(z_+>0\) (the so-called \(q\)-analogue of the real line) \[ \int_{ \mathbb{R}_q} P_m(x) P_n(x) \frac{1}{(cx,dx;q)_\infty }\,d_q x = \delta_{mn} B(c,d;z_-,z_+) (q;q)_n (cd)^n q^{-n^2}, \] with \[ B(c,d;z_-,z_+)=(1-q)z_+\frac{(q;q)_\infty \theta(z_-/z_+,cdz_-z_+;q) }{\theta(cz_-,dz_-,cz_+,dz_+;q)}, \] where \(P_n(x) = d^n V_n^{(c/d)}(cx;q) = (-c)^n q^{-\frac12 n(n-1)} {}_{2}\varphi_{0}({q^{-n},cx}{-}{q,\frac{dq^n}{c}})\), is the Al-Salam-Carlitz polynomials of type II. It is known that the Al-Salam-Carlitz II polynomials correspond to an indeterminate moment problem and the corresponding polynomials \(P_n\) are not dense in the \(L^2\)-space associated to the weight function of the above orthogonality relation. Then, by using the spectral analysis of the corresponding second-order \(q\)-difference operator of which the Al-Salam-Carlitz II polynomials are eigenfunctions, the author obtains an infinite set of functions that complement the Al-Salam-Carlitz II polynomials to an orthogonal basis of the associated \(L^2\)-space. Finally, he also considers the case of the semi-infinite interval \([1/c,\infty)\).