Formula:KLS:09.02:04: Difference between revisions

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Latest revision as of 08:34, 22 December 2019


∑ x = 0 N ( α + 1 ) x ⁢ ( β + δ + 1 ) x ⁢ ( γ + 1 ) x ⁢ ( γ + δ + 1 ) x ⁢ ( ( γ + δ + 3 ) / 2 ) x ( - α + γ + δ + 1 ) x ⁢ ( - β + γ + 1 ) x ⁢ ( ( γ + δ + 1 ) / 2 ) x ⁢ ( δ + 1 ) x ⁢ x ! ⁢ R m ⁡ ( λ ⁢ ( x ) ) ⁢ R n ⁡ ( λ ⁢ ( x ) ) = M ⁢ ( n + α + β + 1 ) n ⁢ ( α + β - γ + 1 ) n ⁢ ( α - δ + 1 ) n ⁢ ( β + 1 ) n ⁢ n ! ( α + β + 2 ) 2 ⁢ n ⁢ ( α + 1 ) n ⁢ ( β + δ + 1 ) n ⁢ ( γ + 1 ) n ⁢ δ m , n superscript subscript 𝑥 0 𝑁 Pochhammer-symbol 𝛼 1 𝑥 Pochhammer-symbol 𝛽 𝛿 1 𝑥 Pochhammer-symbol 𝛾 1 𝑥 Pochhammer-symbol 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛾 𝛿 3 2 𝑥 Pochhammer-symbol 𝛼 𝛾 𝛿 1 𝑥 Pochhammer-symbol 𝛽 𝛾 1 𝑥 Pochhammer-symbol 𝛾 𝛿 1 2 𝑥 Pochhammer-symbol 𝛿 1 𝑥 𝑥 Racah-polynomial-R 𝑚 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 Racah-polynomial-R 𝑛 𝜆 𝑥 𝛼 𝛽 𝛾 𝛿 𝑀 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 𝛽 𝛾 1 𝑛 Pochhammer-symbol 𝛼 𝛿 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Pochhammer-symbol 𝛼 𝛽 2 2 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 𝛿 1 𝑛 Pochhammer-symbol 𝛾 1 𝑛 Kronecker-delta 𝑚 𝑛 {\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)

M = { ( - β ) N ⁢ ( γ + δ + 2 ) N ( - β + γ + 1 ) N ⁢ ( δ + 1 ) N < b r / >   if   α + 1 = - N ( - α + δ ) N ⁢ ( γ + δ + 2 ) N ( - α + γ + δ + 1 ) N ⁢ ( δ + 1 ) N < b r / >   if   β + δ + 1 = - N ( α + β + 2 ) N ⁢ ( - δ ) N ( α - δ + 1 ) N ⁢ ( β + 1 ) N < b r / >   if   γ + 1 = - N 𝑀 cases Pochhammer-symbol 𝛽 𝑁 Pochhammer-symbol 𝛾 𝛿 2 𝑁 Pochhammer-symbol 𝛽 𝛾 1 𝑁 Pochhammer-symbol 𝛿 1 𝑁 fragments b r italic-  if italic-  α 1 N Pochhammer-symbol 𝛼 𝛿 𝑁 Pochhammer-symbol 𝛾 𝛿 2 𝑁 Pochhammer-symbol 𝛼 𝛾 𝛿 1 𝑁 Pochhammer-symbol 𝛿 1 𝑁 fragments b r italic-  if italic-  β δ 1 N Pochhammer-symbol 𝛼 𝛽 2 𝑁 Pochhammer-symbol 𝛿 𝑁 Pochhammer-symbol 𝛼 𝛿 1 𝑁 Pochhammer-symbol 𝛽 1 𝑁 fragments b r italic-  if italic-  γ 1 N {\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.}}} } &
λ ⁢ ( x ) = x ⁢ ( x + γ + δ + 1 ) 𝜆 𝑥 𝑥 𝑥 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\lambda(x)=x(x+\gamma+\delta+1)}}}


Proof

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Symbols List

& : logical and
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
R n subscript 𝑅 𝑛 {\displaystyle{\displaystyle{\displaystyle R_{n}}}}  : Racah polynomial : http://dlmf.nist.gov/18.25#T1.t1.r4
δ m , n subscript 𝛿 𝑚 𝑛 {\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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