DLMF:18.35.E7 (Q6046): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1cdec

Revision as of 13:31, 2 January 2020

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DLMF:18.35.E7
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    Statements

    Defining formula
    ( 1 - z ⁢ e i ⁢ θ ) - λ + i ⁢ τ a , b ⁢ ( θ ) ⁢ ( 1 - z ⁢ e - i ⁢ θ ) - λ - i ⁢ τ a , b ⁢ ( θ ) = ∑ n = 0 ∞ P n ( λ ) ⁡ ( cos ⁡ θ ; a , b ) ⁢ z n , superscript 1 𝑧 superscript 𝑒 imaginary-unit 𝜃 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 superscript 1 𝑧 superscript 𝑒 imaginary-unit 𝜃 𝜆 imaginary-unit subscript 𝜏 𝑎 𝑏 𝜃 superscript subscript 𝑛 0 Pollaczek-polynomial-P 𝜆 𝑛 𝜃 𝑎 𝑏 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1-ze^{\mathrm{i}\theta})^{-\lambda+\mathrm{i}\tau% _{a,b}(\theta)}(1-ze^{-\mathrm{i}\theta})^{-\lambda-\mathrm{i}\tau_{a,b}(% \theta)}=\sum_{n=0}^{\infty}P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)z^{n},}}
    0 references
    DLMF id
    DLMF:18.35.E7
    0 references
    constraint
    0 < θ < π 0 𝜃 𝜋 {\displaystyle{\displaystyle 0<\theta<\pi}}
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    Symbols used
    Pollaczek polynomial
    Defining formula
    P n ( λ ) ⁡ ( x ; a , b ) Pollaczek-polynomial-P 𝜆 𝑛 𝑥 𝑎 𝑏 {\displaystyle{\displaystyle P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};% \NVar{a},\NVar{b}\right)}}
    xml-id
    C18.S35.E4.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q11750
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1dec
    0 references
    Q11726
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1cdec
    0 references
     
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