Formula:KLS:09.04:18

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