Structure relations for the bivariate big q-Jacobi polynomials

The bivariate big q-Jacobi polynomials are defined by [3] 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 ≥ 0; 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]). We give structure relations in the form σ 1 Dq±1,xPn = F ± n,1Pn+1 +G ± n,1Pn +H ± n,1Pn−1, σ 2 Dq±1,yPn = F ± n,2Pn+1 +G ± n,2Pn +H ± n,2Pn−1, where Pn := [Pn,0, Pn,1, . . . , Pn,n] , Pn,k = Pn,k(x, y; a, b, c, d; q), Dq±1,zPn := [Dq±1,zPn,0, Dq±1,zPn,1, . . . , Dq±1,zPn,n] T (z = x, y), Dq±1,x and Dq±1,y are partial q-derivative operators, σ ± i are certain polynomials of total degree 2, and F ± n,i, G ± n,i and H± n,i (i = 1, 2) are tridiagonal matrices of appropriate dimensions. We discuss in full detail the case of the polynomials Pn,k(x, y; a, b, c, 0; q), which are closely related to Dunkl’s bivariate (little) q-Jacobi polynomials [2]. AMS Classification: 33D50, 33C50