https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=199.46.199.230formulasearchengine - User contributions [en]2026-08-04T14:52:07ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Range_ambiguity_resolution&diff=26912Range ambiguity resolution2013-07-31T16:45:57Z<p>199.46.199.230: </p>
<hr />
<div>In mathematics, the '''continuous dual Hahn polynomials''' are a family of [[orthogonal polynomials]] in the [[Askey scheme]] of hypergeometric orthogonal polynomials. They are defined in terms of [[generalized hypergeometric function]]s by <br />
:<math>S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).\ </math><br />
<br />
{{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.<br />
<br />
Closely related polynomials include the [[dual Hahn polynomials]] ''R''<sub>''n''</sub>(''x'';γ,δ,''N''), the [[continuous Hahn polynomials]] ''p''<sub>''n''</sub>(''x'',''a'',''b'', {{overline|''a''}}, {{overline|''b''}}), and the [[Hahn polynomials]]. These polynomials all have ''q''-analogs with an extra parameter ''q'', such as the [[q-Hahn polynomials]] ''Q''<sub>''n''</sub>(''x'';α,β, ''N'';''q''), and so on.<br />
<br />
==Orthogonality==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Recurrence and difference relations==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Rodrigues formula==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Generating function==<br />
{{Empty section|date=September 2011}}<br />
<br />
==Relation to other polynomials==<br />
<br />
*[[Wilson polynomials]] are a generalization of continuous dual Hahn polynomials<br />
<br />
==References==<br />
<br />
*{{Citation | last1=Hahn | first1=Wolfgang | title=Über Orthogonalpolynome, die q-Differenzengleichungen genügen | doi=10.1002/mana.19490020103 | mr=0030647 | year=1949 | journal=Mathematische Nachrichten | issn=0025-584X | volume=2 | pages=4–34}}<br />
*{{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}}<br />
*{{dlmf|id=18.19|title=Hahn Class: Definitions|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}<br />
<br />
[[Category:Special hypergeometric functions]]<br />
[[Category:Orthogonal polynomials]]</div>199.46.199.230