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Q-Hahn polynomials

In mathematics, the 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 by :Q_n(q^{-x};a,b,N;q)={}_3\phi_2\left[\begin{matrix} q^{-n},abq^{n+1},q^{-x}\ aq,q^{-N}\end{matrix} ;q,q\right].

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

\lim_{a \to \infty}Q_{n}(q^{-x};a;p,N|q)=K_{n}^{qtm}(q^{-x};p,N;q)

q-Hahn polynomials→ Hahn polynomials

make the substitution\alpha=q^{\alpha},\beta=q^{\beta} into definition of q-Hahn polynomials, and find the limit q→1, we obtain

:{}_3F_2(-n,\alpha+\beta+n+1,-x,\alpha+1,-N,1)，which is exactly Hahn polynomials.

References

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

orthogonal-polynomials q-analogs special-hypergeometric-functions

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