q-Hahn polynomials

See also: continuous q-Hahn polynomials, dual q-Hahn polynomials, and continuous dual q-Hahn polynomials

In mathematics, the 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 Pochhammer symbol by $Q_n(x;a,b,N;q)={}_3{\varphi }_2[\begin{array}{ccc}{q}^{-}n& ab{q}^n+1& x\\ aq& q^{-}N\end{array};q,q]$

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

$\underset{a\to \mathrm{\infty }}{\lim }{Q}_n(q^{-}x;a;p,N|q)=K_n^{qtm}(q^{-}x;p,N;q)$

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

： $3_{}{F}_2([-n,\alpha +\beta +n+1,-x],[\alpha +1,-N],1)$，which is exactly Hahn polynomials.

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), http://dlmf.nist.gov/18, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248
• Costas-Santos, R.S.; Sánchez-Lara, J.F. (September 2011). "Orthogonality of q-polynomials for non-standard parameters". Journal of Approximation Theory. 163 (9): 1246–1268. arXiv:1002.4657. doi:10.1016/j.jat.2011.04.005.