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DLMF:18.23.E7 - MaRDI portal

DLMF:18.23.E7 (Q6604)

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DLMF:18.23.E7
No description defined

    Statements

    DLMF defining formula
    ( 1 - e i ⁢ ϕ ⁢ z ) - λ + i ⁢ x ⁢ ( 1 - e - i ⁢ ϕ ⁢ z ) - λ - i ⁢ x = ∑ n = 0 ∞ P n ( λ ) ⁡ ( x ; ϕ ) ⁢ z n , superscript 1 superscript 𝑒 imaginary-unit italic-ϕ 𝑧 𝜆 imaginary-unit 𝑥 superscript 1 superscript 𝑒 imaginary-unit italic-ϕ 𝑧 𝜆 imaginary-unit 𝑥 superscript subscript 𝑛 0 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1-e^{\mathrm{i}\phi}z)^{-\lambda+\mathrm{i}x}(1-e% ^{-\mathrm{i}\phi}z)^{-\lambda-\mathrm{i}x}=\sum_{n=0}^{\infty}P^{(\lambda)}_{% n}\left(x;\phi\right)z^{n},}}
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    Symbols used
    Meixner–Pollaczek polynomial
    DLMF defining formula
    P n ( λ ) ⁡ ( x ; ϕ ) Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ {\displaystyle{\displaystyle P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};% \NVar{\phi}\right)}}
    xml-id
    C18.S19.Px3.p1.m2adec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1fdec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1fdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1fdec
    0 references

    Identifiers

     
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