(1, 1)-q-coherent pairs

Abstract

In this paper, we introduce the concept of (1, 1)-q-coherent pair of linear functionals $(\mathcal{U},\mathcal{V})$ as the q-analogue to the generalized coherent pair studied by Delgado and Marcellán in (Methods Appl Anal 11(2):273–266, 2004). This means that their corresponding sequences of monic orthogonal polynomials {P _n (x)} _n ≥ 0 and {R _n (x)} _n ≥ 0 satisfy

$$ \frac{\left(D_qP_{n+1}\right)(x)}{[n+1]_q} + a_{n}\frac{\left(D_qP_{n}\right)(x)}{[n]_q} = R_{n}(x) + b_{\!n}R_{n-1}(x) \,, \quad\, a_{n}\neq0,\,\, n\geq1, $$

$[n]_q=\frac{q^n-1}{q-1}$, 0 < q < 1. We prove that if a pair of regular linear functionals $(\mathcal{U},\mathcal{V})$ is a (1, 1)-q-coherent pair, then at least one of them must be q-semiclassical of class at most 1, and these functionals are related by an expression $\sigma(x)\mathcal{U}=\rho(x)\mathcal{V}$ where σ(x) and ρ(x) are polynomials of degrees ≤ 3 and 1, respectively. Finally, the q-classical case is studied.