Dual q-Hahn polynomials

{{Short description|Family of hypergeometric orthogonal polynomials}}

In mathematics, the dual q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. {{harvs|txt | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010|loc=14}} give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions.

$R_n(q^{-x}+\gamma\delta q^{x+1},\gamma,\delta,N|q)={}_3\phi_2\left[\begin{matrix}
q^{-n},q^{-x},\gamma\delta q^{x+1}\\
\gamma q,q^{-N}\end{matrix}
;q,q\right],\quad n=0,1,2,...,N$

References

{{Citation | last1=Gasper | first1=George | authorlink1 = George Gasper | last2=Rahman | first2=Mizan | authorlink2 = Mizan Rahman | title=Basic hypergeometric series | publisher=Cambridge University Press | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | mr=2128719 | year=2004 | volume=96}}
{{Citation | last1=Koekoek | first1=Roelof | last2=Lesky | first2=Peter A. | last3=Swarttouw | first3=René F. | title=Hypergeometric orthogonal polynomials and their q-analogues | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-642-05013-8 | doi=10.1007/978-3-642-05014-5 | mr=2656096 | year=2010}}
{{dlmf|id=18|title=Chapter 18 Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
{{cite journal|last1=Costas-Santos|first1=R.S.|last2=Sánchez-Lara|first2=J.F.|title=Orthogonality of q-polynomials for non-standard parameters|journal=Journal of Approximation Theory|date=September 2011|volume=163|issue=9|pages=1246–1268|doi=10.1016/j.jat.2011.04.005|arxiv=1002.4657|s2cid=115178147 }}
{{cite thesis |last=Sadjang |first=Patrick Njionou |title=Moments of Classical Orthogonal Polynomials |type=Ph.D. |publisher=Universität Kassel |citeseerx=10.1.1.643.3896 }}

Category:Orthogonal polynomials

Category:Q-analogs

Category:Special hypergeometric functions