Some \(q\)-orthogonal polynomials and related Hankel determinants. (Q1415028)

Put, as usual, \[ \left(A;q\right)_n =\prod_{j=0}^{n-1}\left(1-Aq^j\right),\quad n>0,\quad \left(A;q\right)_0=1; \] \[ \left[{n\atop j}\right]={(q;q)_n\over (q;q)_j(q;q)_{n-j}}, \] and define a polynomial sequence by: \[ S_0(x)=1;\quad S_0S_N(x)=\sum_{j=0}^{N-1} \left[{N-1\atop j}\right] \left(aq^{j+3};q\right)_{N-1-j}\over \left(bq^{N+1+j};q\right)_{N-1-j}\tau_{j+1}(x)\times q^{\begin{pmatrix} {N-1-j\atop 2}\end{pmatrix}}(-1)^{N-1-j},\quad N>0, \] where \[ \tau_n(x)=\sum_{s=0}^{n} {(aq;q)_{n-s}\over (bq;q)_{n-s} }x^s. \] In this paper, the authors prove that the polynomials \(S_N(x)\) are orthogonal with respect to a complex distribution defined by \[ L\left(h(z)\right)={1\over 2\pi i}\int_{\mathcal C}h(z) \left({f\left(1/z\right)\over z}\right)dz, \] where \[ f(t)={1\over \sum_{n=0}^{\infty} {t^n(a;q)_{n+1}\over (b;q)_{n+1} }}, \] and the path of integration \({\mathcal C}\) is a simple closed curve encircling the origin which lies outside all singularities of the integrand. They, also, express explicitly the coefficients \(B_N\) and \(C_N\) in the three-term recurrence relation: \[ S_{N+1}(x)=\left(x+B_N\right)S_N(x)-C_NS_{N-1}(x). \] Finally, they mention three interesting special cases of this results.