Formula:KLS:09.05:29

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∑ n = 0 N ( 2 ⁢ n + α + β + 1 ) ⁢ ( α + 1 ) n ⁢ ( - N ) n ⁢ N ! ( - 1 ) n ⁢ ( n + α + β + 1 ) N + 1 ⁢ ( β + 1 ) n ⁢ n ! ⁢ Q n ⁡ ( x ; α , β , N ) ⁢ Q n ⁡ ( y ; α , β , N ) = δ x , y ( α + x x ) ⁢ ( β + N - x N - x ) superscript subscript 𝑛 0 𝑁 2 𝑛 𝛼 𝛽 1 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝑁 𝑛 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑁 1 Pochhammer-symbol 𝛽 1 𝑛 𝑛 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑦 𝛼 𝛽 𝑁 Kronecker-delta 𝑥 𝑦 binomial 𝛼 𝑥 𝑥 binomial 𝛽 𝑁 𝑥 𝑁 𝑥 {\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{N}\frac{(2n+\alpha+\beta% +1){\left(\alpha+1\right)_{n}}{\left(-N\right)_{n}}N!}{(-1)^{n}{\left(n+\alpha% +\beta+1\right)_{N+1}}{\left(\beta+1\right)_{n}}n!}Q_{n}\!\left(x;\alpha,\beta% ,N\right)Q_{n}\!\left(y;\alpha,\beta,N\right){}=\frac{\delta_{x,y}}{\dbinom{% \alpha+x}{x}\dbinom{\beta+N-x}{N-x}}}}}

Constraint(s)

x , y ∈ { 0 , 1 , 2 , … , N } 𝑥 𝑦 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle x,y\in\{0,1,2,\ldots,N\}}}}


Proof

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Symbols List

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
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3
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
∈ {\displaystyle{\displaystyle{\displaystyle\in}}}  : element of : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r9

Bibliography

Equation in Section 9.5 of KLS.

URL links

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