Two-variable orthogonal polynomials of big q-Jacobi type

A four-parameter family of orthogonal polynomials in two variables is defined by Pn,k(x, y; a, b, c, d; q) :=Pn−k(y; a, bcq , dq; q) y(dq/y; q)k Pk (x/y; c, b, d/y; q) (n ∈ N; k = 0, 1, . . . , n), where q ∈ (0, 1), 0 < aq, bq, cq < 1, d < 0, and Pm(t;α, β, γ; q) are univariate big q-Jacobi polynomials, Pm(t;α, β, γ; q) := 3φ2 ( q−m, αβq, t αq, γq ∣∣∣∣ q; q) (m ≥ 0) (see, e.g., [1, Section 7.3]). These polynomials form an extension of Dunkl’s bivariate (little) q-Jacobi polynomials [2]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation. We give some basic properties of the new polynomials. First, we show that they form an orthogonal system with respect to the linear functional u defined by 〈u, p〉 := ∫ aq