DLMF:15.10.E31 (Q5147)

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DLMF:15.10.E31
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    Statements

    Defining formula
    w 2 ⁢ ( z ) = e ( b - c + 1 ) ⁢ π ⁢ i ⁢ Γ ⁡ ( 2 - c ) ⁢ Γ ⁡ ( a ) Γ ⁡ ( a + b - c + 1 ) ⁢ Γ ⁡ ( 1 - b ) ⁢ w 3 ⁢ ( z ) + e ( b - c ) ⁢ π ⁢ i ⁢ Γ ⁡ ( 2 - c ) ⁢ Γ ⁡ ( a ) Γ ⁡ ( a - b + 1 ) ⁢ Γ ⁡ ( b - c + 1 ) ⁢ w 5 ⁢ ( z ) , subscript 𝑤 2 𝑧 superscript 𝑒 𝑏 𝑐 1 𝜋 imaginary-unit Euler-Gamma 2 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 𝑐 1 Euler-Gamma 1 𝑏 subscript 𝑤 3 𝑧 superscript 𝑒 𝑏 𝑐 𝜋 imaginary-unit Euler-Gamma 2 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 𝑏 𝑐 1 subscript 𝑤 5 𝑧 {\displaystyle{\displaystyle w_{2}(z)=e^{(b-c+1)\pi\mathrm{i}}\frac{\Gamma% \left(2-c\right)\Gamma\left(a\right)}{\Gamma\left(a+b-c+1\right)\Gamma\left(1-% b\right)}w_{3}(z)+e^{(b-c)\pi\mathrm{i}}\frac{\Gamma\left(2-c\right)\Gamma% \left(a\right)}{\Gamma\left(a-b+1\right)\Gamma\left(b-c+1\right)}w_{5}(z),}}
    0 references
    DLMF id
    DLMF:15.10.E31
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2andec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2akdec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2akdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1addec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1addec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1addec
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    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1aadec
    0 references
    Q11667
    Defining formula
    w m ⁢ ( z ) subscript 𝑤 𝑚 𝑧 {\displaystyle{\displaystyle w_{m}(z)}}
    xml-id
    C15.S10.XMD2.m1tdec
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