Formula:KLS:09.04:16

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δ ⁢ [ ω ⁢ ( x ; a , b , c , d ) ⁢ p n ⁡ ( x ; a , b , c , d ) ] δ ⁢ x = - ( n + 1 ) ⁢ ω ⁢ ( x ; a - 1 2 , b - 1 2 ⁢ c - 1 2 , d - 1 2 ) ⁢ p n + 1 ⁡ ( x ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝛿 delimited-[] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝛿 𝑥 𝑛 1 𝜔 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 continuous-Hahn-polynomial 𝑛 1 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c,d)p% _{n}\!\left(x;a,b,c,d\right)\right]}{\delta x}{}=-(n+1)\omega(x;a-\textstyle% \frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1% }{2}){}p_{n+1}\!\left(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-% \textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}}

Substitution(s)

ω ⁢ ( x ; a , b , c , d ) = Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c - i ⁢ x ) ⁢ Γ ⁡ ( d - i ⁢ x ) 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)=\Gamma\left(a+% \mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c-\mathrm{i}x% \right)\Gamma\left(d-\mathrm{i}x\right)}}}


Proof

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Symbols List

p n subscript 𝑝 𝑛 {\displaystyle{\displaystyle{\displaystyle p_{n}}}}  : continuous Hahn polynomial : http://dlmf.nist.gov/18.19#P2.p1
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.4 of KLS.

URL links

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