Formula:KLS:09.02:04

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formula in Racah

Formula:KLS:09.02:05 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{N}\frac{{\left(\alpha+1\right)_{x}}{\left(\beta+\delta+1\right)_{x}}{\left(\gamma+1\right)_{x}}{\left(\gamma+\delta+1\right)_{x}}{\left((\gamma+\delta+3)/2\right)_{x}}}{{\left(-\alpha+\gamma+\delta+1\right)_{x}}{\left(-\beta+\gamma+1\right)_{x}}{\left((\gamma+\delta+1)/2\right)_{x}}{\left(\delta+1\right)_{x}}x!}{}R_{m}\!\left(\lambda(x)\right)R_{n}\!\left(\lambda(x)\right){}=M\frac{{\left(n+\alpha+\beta+1\right)_{n}}{\left(\alpha+\beta-\gamma+1\right)_{n}}{\left(\alpha-\delta+1\right)_{n}}{\left(\beta+1\right)_{n}}n!}{{\left(\alpha+\beta+2\right)_{2n}}{\left(\alpha+1\right)_{n}}{\left(\beta+\delta+1\right)_{n}}{\left(\gamma+1\right)_{n}}}\,\delta_{m,n}}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle M=\left\{\begin{array}[]{ll}\displaystyle\frac{{\left(-\beta\right)_{N}}{\left(\gamma+\delta+2\right)_{N}}}{{\left(-\beta+\gamma+1\right)_{N}}{\left(\delta+1\right)_{N}}}&<br/>\quad\textrm{if}\quad\alpha+1=-N\\ \displaystyle\frac{{\left(-\alpha+\delta\right)_{N}}{\left(\gamma+\delta+2\right)_{N}}}{{\left(-\alpha+\gamma+\delta+1\right)_{N}}{\left(\delta+1\right)_{N}}}&<br/>\quad\textrm{if}\quad\beta+\delta+1=-N\\ \displaystyle\frac{{\left(\alpha+\beta+2\right)_{N}}{\left(-\delta\right)_{N}}}{{\left(\alpha-\delta+1\right)_{N}}{\left(\beta+1\right)_{N}}}&<br/>\quad\textrm{if}\quad\gamma+1=-N\end{array}\right.}}}$ } &
${\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}$ : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
${\displaystyle{\displaystyle{\displaystyle R_{n}}}}$ : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 9.2 of KLS.

URL links

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<< Formula:KLS:09.02:03

formula in Racah

Formula:KLS:09.02:05 >>