Formula:KLS:09.05:16: Difference between revisions

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Latest revision as of 08:34, 22 December 2019


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