Formula:KLS:09.05:18: Difference between revisions

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ω ⁢ ( x ; α , β , N ) ⁢ Q n ⁡ ( x ; α , β , N ) = ( - 1 ) n ⁢ ( β + 1 ) n ( - N ) n ⁢ ∇ n ⁡ [ ω ⁢ ( x ; α + n , β + n , N - n ) ] 𝜔 𝑥 𝛼 𝛽 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝛼 𝛽 𝑁 superscript 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Pochhammer-symbol 𝑁 𝑛 superscript ∇ 𝑛 𝜔 𝑥 𝛼 𝑛 𝛽 𝑛 𝑁 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)Q_{n}\!\left% (x;\alpha,\beta,N\right){}=\frac{(-1)^{n}{\left(\beta+1\right)_{n}}}{{\left(-N% \right)_{n}}}\nabla^{n}\left[\omega(x;\alpha+n,\beta+n,N-n)\right]}}}

Substitution(s)

ω ⁢ ( x ; α , β , N ) = \binomial ⁢ α + x ⁢ x ⁢ \binomial ⁢ β + N - x ⁢ N - x 𝜔 𝑥 𝛼 𝛽 𝑁 \binomial 𝛼 𝑥 𝑥 \binomial 𝛽 𝑁 𝑥 𝑁 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;\alpha,\beta,N)=\binomial{% \alpha+x}{x}\binomial{\beta+N-x}{N-x}}}}


Proof

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Symbols List

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
( 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

Bibliography

Equation in Section 9.5 of KLS.

URL links

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