Formula:KLS:09.04:27

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p n ⁡ ( x ; a , a + 1 2 , a , a + 1 2 ) = ( 2 ⁢ a ) n ⁢ ( 2 ⁢ a + 1 2 ) n ( 4 ⁢ a ) n ⁢ P n ( 2 ⁢ a ) ⁡ ( 2 ⁢ x ; 1 2 ⁢ π ) continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑎 1 2 𝑎 𝑎 1 2 Pochhammer-symbol 2 𝑎 𝑛 Pochhammer-symbol 2 𝑎 1 2 𝑛 Pochhammer-symbol 4 𝑎 𝑛 Meixner-Pollaczek-polynomial-P 2 𝑎 𝑛 2 𝑥 1 2 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,a+\frac{1}{2},a,a+% \frac{1}{2}\right)=\frac{{\left(2a\right)_{n}}{\left(2a+\frac{1}{2}\right)_{n}% }}{{\left(4a\right)_{n}}}P^{(2a)}_{n}\!\left(2x;\frac{1}{2}\pi\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
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
P n ( α ) subscript superscript 𝑃 𝛼 𝑛 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha)}_{n}}}}  : Meixner-Pollaczek polynomial : http://dlmf.nist.gov/18.19#P3.p1

Bibliography

Equation in Section 9.4 of KLS.

URL links

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