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Little q-Jacobi polynomials

In mathematics, the little q-Jacobi polynomials p**n(x;a,b;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties.

Definition

The little q-Jacobi polynomials are given in terms of basic hypergeometric functions by :\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)

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.

[[File:LITTLE Q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS ABS COMPLEX 3D MAPLE PLOT]]	[[File:LITTLE Q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS IM COMPLEX 3D MAPLE PLOT]]	[[File:LITTLE Q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS RE COMPLEX 3D MAPLE PLOT]]

[[File:LITTLE Q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS ABS DENSITY MAPLE PLOT]]	[[File:LITTLE Q-JACOBI POLYNOMIALS IM DENSITY MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS IM DENSITY MAPLE PLOT]]	[[File:LITTLE Q-JACOBI POLYNOMIALS RE DENSITY MAPLE PLOT.gif	thumb	LITTLE q-JACOBI POLYNOMIALS RE DENSITY MAPLE PLOT]]

References

(1949). "Über Orthogonalpolynome, die q-Differenzengleichungen genügen". [[Mathematische Nachrichten]].
{{dlmf. Tom H.. Koornwinder

Wikipedia Source

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orthogonal-polynomials q-analogs special-hypergeometric-functions

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