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 by

[math]\displaystyle{ 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]. }[/math]

Relation to other polynomials

q-Hahn polynomials→ Quantum q-Krawtchouk polynomials：

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

q-Hahn polynomials→ Hahn polynomials

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

[math]\displaystyle{ {}_3F_2(-n,\alpha+\beta+n+1,-x,\alpha+1,-N,1) }[/math]，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, ISBN 978-0-521-83357-8 
• 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 
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18 
• 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. doi:10.1016/j.jat.2011.04.005.