Formula:KLS:09.01:11

<< Formula:KLS:09.01:10

formula in Wilson

Formula:KLS:09.01:15 >>

${\displaystyle{\displaystyle{\displaystyle n(n+a+b+c+d-1)y(x){}=B(x)y(x+\mathrm{i})-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\mathrm{i})}}}$

Substitution(s)

${\displaystyle{\displaystyle{\displaystyle D(x)=\frac{(a+\mathrm{i}x)(b+\mathrm{i}x)(c+\mathrm{i}x)(d+\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x+1)}}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(a-\mathrm{i}x)(b-\mathrm{i}x)(c-\mathrm{i}x)(d-\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x-1)}}}}$ &

${\displaystyle{\displaystyle{\displaystyle y(x)=W_{n}\!\left(x^{2};a,b,c,d\right)}}}$

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle W_{n}}}}$ : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2

Bibliography

Equation in Section 9.1 of KLS.

URL links

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<< Formula:KLS:09.01:10

formula in Wilson

Formula:KLS:09.01:15 >>