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Continuous q-Hahn polynomials

Hypergeometric orthogonal polynomials

Summary

Hypergeometric orthogonal polynomials

In mathematics, the continuous q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. 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}(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)

Gallery

[[File:CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT.gif	thumb	CONTINUOUS q hahn ABS COMPLEX3D Maple PLOT]]	[[File:CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT.gif	thumb	CONTINUOUS q hahn IIM COMPLEX3D Maple PLOT]]	[[File:CONTINUOUS q hahn RE COMPLEX3D Maple PLOT.gif	thumb	CONTINUOUS q hahn RE COMPLEX3D Maple PLOT]]

[[File:CONTINUOUS q hahn ABS densitu Maple PLOT.gif	thumb	CONTINUOUS q hahn ABS density Maple PLOT]]	[[File:CONTINUOUS q hahn im densitu Maple PLOT.gif	thumb	CONTINUOUS q hahn im density Maple PLOT]]	[[File:CONTINUOUS q hahn RE densitu Maple PLOT.gif	thumb	CONTINUOUS q hahn RE density Maple PLOT]]

References

Roelof p433, Springer 2010

Wikipedia Source

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orthogonal-polynomials q-analogs special-hypergeometric-functions

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