Formula:KLS:09.05:02

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

Proof

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Symbols List

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( n k ) binomial 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
Q n subscript 𝑄 𝑛 {\displaystyle{\displaystyle{\displaystyle Q_{n}}}}  : Hahn polynomial : http://dlmf.nist.gov/18.19#T1.t1.r3
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
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 9.5 of KLS.

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