Formula:KLS:09.01:05: Difference between revisions

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Latest revision as of 08:34, 22 December 2019


- ( a 2 + x 2 ) ⁢ W ~ n ⁡ ( x 2 ) = A n ⁢ W ~ n + 1 ⁡ ( x 2 ) - ( A n + C n ) ⁢ W ~ n ⁡ ( x 2 ) + C n ⁢ W ~ n - 1 ⁡ ( x 2 ) superscript 𝑎 2 superscript 𝑥 2 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐶 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle-\left(a^{2}+x^{2}\right){\tilde{W}}% _{n}\!\left(x^{2}\right)=A_{n}{\tilde{W}}_{n+1}\!\left(x^{2}\right)-\left(A_{n% }+C_{n}\right){\tilde{W}}_{n}\!\left(x^{2}\right)+C_{n}{\tilde{W}}_{n-1}\!% \left(x^{2}\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{\tilde{W}}_{n}}}}  : normalized Wilson polynomial W ~ ~ 𝑊 {\displaystyle{\displaystyle{\displaystyle{\tilde{W}}}}}  : http://drmf.wmflabs.org/wiki/Definition:normWilsonWtilde

Bibliography

Equation in Section 9.1 of KLS.

URL links

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