Little q-Jacobi polynomials

From HandWiki

In mathematics, the little q-Jacobi polynomials pn(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by (Hahn 1949). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The little q-Jacobi polynomials are given in terms of basic hypergeometric functions by

[math]\displaystyle{ \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq) }[/math]

Gallery

The following are a set of animation plots for Little q-Jacobi polynomials, with varying q; three density plots of imaginary, real and modulus in complex space; three set of complex 3D plots of imaginary, real and modulus of the said polynomials.

LITTLE q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT
LITTLE q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT
LITTLE q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT
LITTLE q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT
LITTLE q-JACOBI POLYNOMIALS IM DENSITY MAPLE PLOT
LITTLE q-JACOBI POLYNOMIALS RE DENSITY MAPLE PLOT

References