Formula:KLS:09.01:17

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( a - 1 2 - i ⁢ x ) ⁢ ( b - 1 2 - i ⁢ x ) ⁢ ( c - 1 2 - i ⁢ x ) ⁢ ( d - 1 2 - i ⁢ x ) ⁢ W n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c , d ) - ( a - 1 2 + i ⁢ x ) ⁢ ( b - 1 2 + i ⁢ x ) ⁢ ( c - 1 2 + i ⁢ x ) ⁢ ( d - 1 2 + i ⁢ x ) ⁢ W n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c , d ) = - 2 ⁢ i ⁢ x ⁢ W n + 1 ⁡ ( x 2 ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 𝑑 1 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 𝑑 1 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle(a-\textstyle\frac{1}{2}-\mathrm{i}x% )(b-\textstyle\frac{1}{2}-\mathrm{i}x)(c-\textstyle\frac{1}{2}-\mathrm{i}x)(d-% \textstyle\frac{1}{2}-\mathrm{i}x)W_{n}\!\left((x+\textstyle\frac{1}{2}\mathrm% {i})^{2};a,b,c,d\right){}-(a-\textstyle\frac{1}{2}+\mathrm{i}x)(b-\textstyle% \frac{1}{2}+\mathrm{i}x)(c-\textstyle\frac{1}{2}+\mathrm{i}x)(d-\textstyle% \frac{1}{2}+\mathrm{i}x)W_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{i})^{2};a% ,b,c,d\right){}=-2\mathrm{i}xW_{n+1}\!\left(x^{2};a-\textstyle\frac{1}{2},b-% \textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}}

Proof

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Symbols List

i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
W n subscript 𝑊 𝑛 {\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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