Formula:KLS:09.07:09

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

Proof

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Symbols List

e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
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 9.7 of KLS.

URL links

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