Formula:KLS:09.01:09

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x ⁢ W ^ n ⁡ ( x ) = W ^ n + 1 ⁡ ( x ) + ( A n + C n - a 2 ) ⁢ W ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ W ^ n - 1 ⁡ ( x ) 𝑥 Wilson-polynomial-monic 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-monic 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 subscript 𝐶 𝑛 superscript 𝑎 2 Wilson-polynomial-monic 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Wilson-polynomial-monic 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle x{\widehat{W}}_{n}\!\left(x\right)=% {\widehat{W}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}-a^{2}){\widehat{W}}_{n}\!% \left(x\right)+A_{n-1}C_{n}{\widehat{W}}_{n-1}\!\left(x\right)}}}

Substitution(s)

C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ⁢ ( n + c + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 𝑛 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+b+c-1)(n+b+d-1)(n+c% +d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}}} &
A n = ( n + a + b + c + d - 1 ) ⁢ ( n + a + b ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑏 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle A_{n}=\frac{(n+a+b+c+d-1)(n+a+b)(n+% a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}}


Proof

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Symbols List

& : logical and
W ^ n subscript ^ 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle{\widehat{W}}_{n}}}}  : monic Wilson polynomial : http://drmf.wmflabs.org/wiki/Definition:monicWilson

Bibliography

Equation in Section 9.1 of KLS.

URL links

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