Asymptotics for multiple $q$-orthogonal polynomials from the RHP

Abstract: We deduce the asymptotic behaviour of a broad class of multiple $q$-orthogonal polynomials as their degree tends to infinity. We achieve this by rephrasing multiple $q$-orthogonal polynomials as part of a solution to a Riemann Hilbert Problem (RHP). In particular, we study multiple $q$-orthogonal polynomials of the first kind (see [12]), which are Type II orthogonal polynomials with weights given by \begin{equation} w_1(x) = x^\alpha \omega(x)d_qx,\qquad w_2(x) = x^\beta \omega(x)d_qx, \nonumber \end{equation} which satisfy the constraint \begin{equation}\nonumber |\omega(q^{2n})-1| = \mathcal{O}(q^{2n}), \end{equation} as $n\to \infty$. Using $q$-calculus we obtain detailed asymptotics for these polynomials from the RHP. This class of polynomials studied was chosen in part to their connection to the work of [11,12], concerning the irrationality of $\zeta_q(1)$ and $\zeta_q(2)$. To conduct our asymptotic analysis we will require the following added restrictions on $w_1(x)$ and $w_2(x)$: $\alpha \notin \mathbb{Z}$, $\beta \notin \mathbb{Z}$ and $\alpha \neq \beta \mod \mathbb{Z}$. These restrictions are necessary for the asymptotic analysis but not the statement of multiple $q$-orthogonal polynomials as solutions to a RHP. The author wishes to extend special thanks to Prof. Walter Van Assche, who motivated this studied and provided valuable discussion.