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Continuous q-Hahn polynomials

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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 i n u ( a b e 2 i u , a c , a d ; q ) n 4 ϕ 3 ( q n , a b c d q n 1 , a e i ( t + 2 u ) , a e i t ; a b e 2 i u , a c , a d ; q ; q ) {\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)}

x = cos ( t + u ) {\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

References

  1. Roelof p433, Springer 2010
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