(1, 1)‐q‐coherent pairs

In this paper, we introduce the concept of (1, 1)‐q‐coherent pair of linear functionals $(𝒰,𝒱)$ as the q‐analogue to the generalized coherent pair studied by Delgado and Marcellán (2004). This means that their corresponding sequences of monic orthogonal polynomials {P n (x)} n ≥ 0 and {R n (x)} n ≥ 0 satisfy $$\frac{\left(D_q{P}_{n+1}\right)(x)}{[n+1{]}_q}+a_n\frac{\left(D_q{P}_n\right)(x)}{[n{]}_q}=R_n(x)+b_n{R}_{n-<!--\; -\; -->1}(x),a_n\ne <!--\; ?\; -->0,n\ge <!--\; =\; -->1,$$ , 0 < q < 1. We prove that if a pair of regular linear functionals $(𝒰,𝒱)$ 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 $\sigma (x)𝒰=\mathrm{ho}(x)𝒱$ where σ(x) and ρ(x) are polynomials of degrees ≤ 3 and 1, respectively. Finally, the q‐classical case is studied.