https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=217.184.64.0%2F24formulasearchengine - User contributions [en]2026-08-01T21:09:57ZUser contributionsMediaWiki 1.47.0-wmf.7https://en.formulasearchengine.com/w/index.php?title=Charlier_polynomials&diff=22970Charlier polynomials2014-01-05T16:56:21Z<p>217.184.64.130: /* References */ putting in the umlauts.</p>
<hr />
<div>In [[mathematics]], '''Racah polynomials''' are [[orthogonal polynomials]] named after [[Giulio Racah]], as their orthogonality relations are equivalent to his orthogonality relations for [[Racah coefficient]]s.<br />
<br />
The Racah polynomials were first defined by {{harvtxt|Wilson|1978}} and are given by<br />
:<math>p_n(x(x+\gamma+\delta+1)) = {}_4F_3\left[\begin{matrix} -n &n+\alpha+\beta+1&-x&x+\gamma+\delta+1\\<br />
\alpha+1&\gamma+1&\beta+\delta+1\\ \end{matrix};1\right].</math><br />
<br />
{{harvtxt|Askey|Wilson|1979}} introduced the ''q''-Racah polynomials defined in terms of [[basic hypergeometric function]]s by<br />
:<math>p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\<br />
aq&bdq&cq\\ \end{matrix};q;q\right].</math><br />
They are sometimes given with changes of variables as<br />
:<math>W_n(x;a,b,c,N;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&cq^{x-n}\\<br />
aq&bcq&q^{-N}\\ \end{matrix};q;q\right].</math><br />
<br />
==References==<br />
*{{Citation | last1=Askey | first1=Richard | last2=Wilson | first2=James | title=A set of orthogonal polynomials that generalize the Racah coefficients or 6-j symbols | doi=10.1137/0510092 | id={{MathSciNet | id = 541097}} | year=1979 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=10 | issue=5 | pages=1008–1016}}<br />
*{{citation|first=J.|last= Wilson|title= Hypergeometric series recurrence relations and some new orthogonal functions|series= Ph.D. thesis|publisher= Univ. Wisconsin, Madison|year= 1978}}<br />
[[Category:Orthogonal polynomials]]<br />
<br />
{{algebra-stub}}</div>217.184.64.130