Continuous q-Hahn polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ^[1]

${\displaystyle p_{n}(x;a,b,c,d|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_{n}{}_{4}\phi _{3}(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)}$

${\displaystyle x=\cos(t+u)}$

Gallery

CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT

CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT

CONTINUOUS q hahn RE COMPLEX3D Maple PLOT

CONTINUOUS q hahn ABS density Maple PLOT

CONTINUOUS q hahn im density Maple PLOT

CONTINUOUS q hahn RE density Maple PLOT