Formula:KLS:09.07:07

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P n ( λ ) ⁡ ( x + 1 2 ⁢ i ; ϕ ) - P n ( λ ) ⁡ ( x - 1 2 ⁢ i ; ϕ ) = ( e i ⁢ ϕ - e - i ⁢ ϕ ) ⁢ P n - 1 ( λ + 1 2 ) ⁡ ( x ; ϕ ) Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 1 2 imaginary-unit italic-ϕ imaginary-unit italic-ϕ imaginary-unit italic-ϕ Meixner-Pollaczek-polynomial-P 𝜆 1 2 𝑛 1 𝑥 italic-ϕ {\displaystyle{\displaystyle{\displaystyle P^{(\lambda)}_{n}\!\left(x+% \textstyle\frac{1}{2}\mathrm{i};\phi\right)-P^{(\lambda)}_{n}\!\left(x-% \textstyle\frac{1}{2}\mathrm{i};\phi\right)=({\mathrm{e}^{\mathrm{i}\phi}}-{% \mathrm{e}^{-\mathrm{i}\phi}})P^{(\lambda+\frac{1}{2})}_{n-1}\!\left(x;\phi% \right)}}}

Proof

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Symbols List

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
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11

Bibliography

Equation in Section 9.7 of KLS.

URL links

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