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

$p_n(x;a,b,c,d|q)=a^{-n}{e}^{-inu}(ab{e}^{2iu},ac,ad;q{)}_n{}_4{\varphi }_3(q^{-n},abcd{q}^{n-1},a{e}^{i(t+2u)},a{e}^{-it};ab{e}^{2iu},ac,ad;q;q)$

$x=\cos (t+u)$

Gallery

File:CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT.gif CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT

File:CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT.gif CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT

File:CONTINUOUS q hahn RE COMPLEX3D Maple PLOT.gif CONTINUOUS q hahn RE COMPLEX3D Maple PLOT

File:CONTINUOUS q hahn ABS densitu Maple PLOT.gif CONTINUOUS q hahn ABS density Maple PLOT

File:CONTINUOUS q hahn im densitu Maple PLOT.gif CONTINUOUS q hahn im density Maple PLOT

File:CONTINUOUS q hahn RE densitu Maple PLOT.gif CONTINUOUS q hahn RE density Maple PLOT

References

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