DLMF:16.11.E8 (Q5242): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: DLMF:16.11.E5 / rank
 
Normal rank
Property / Symbols used: DLMF:16.11.E5 / qualifier
 
Defining formula:

e k , m subscript 𝑒 𝑘 𝑚 {\displaystyle{\displaystyle e_{k,m}}}

e_{k,m}
Property / Symbols used: DLMF:16.11.E5 / qualifier
 
xml-id: C16.S11.XMD5.m1bdec

Latest revision as of 14:39, 2 January 2020

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DLMF:16.11.E8
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    Statements

    Defining formula
    ( ∏ ℓ = 1 q - 1 Γ ⁡ ( a ℓ ) / ∏ ℓ = 1 q Γ ⁡ ( b ℓ ) ) ⁢ F q q - 1 ⁡ ( a 1 , … , a q - 1 b 1 , … , b q ; - z ) ∼ H q - 1 , q ⁢ ( z ) + E q - 1 , q ⁢ ( z ⁢ e - π ⁢ i ) + E q - 1 , q ⁢ ( z ⁢ e π ⁢ i ) , asymptotic-expansion superscript subscript product ℓ 1 𝑞 1 Euler-Gamma subscript 𝑎 ℓ superscript subscript product ℓ 1 𝑞 Euler-Gamma subscript 𝑏 ℓ Gauss-hypergeometric-pFq 𝑞 1 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑞 1 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 subscript 𝐻 𝑞 1 𝑞 𝑧 subscript 𝐸 𝑞 1 𝑞 𝑧 superscript 𝑒 𝜋 imaginary-unit subscript 𝐸 𝑞 1 𝑞 𝑧 superscript 𝑒 𝜋 imaginary-unit {\displaystyle{\displaystyle\left({\textstyle\ifrac{\prod\limits_{\ell=1}^{q-1% }\Gamma\left(a_{\ell}\right)}{\prod\limits_{\ell=1}^{q}\Gamma\left(b_{\ell}% \right)}}\right){{}_{q-1}F_{q}}\left({a_{1},\dots,a_{q-1}\atop b_{1},\dots,b_{% q}};-z\right)\sim H_{q-1,q}(z)+E_{q-1,q}(ze^{-\pi\mathrm{i}})+E_{q-1,q}(ze^{% \pi\mathrm{i}}),}}
    0 references
    DLMF id
    DLMF:16.11.E8
    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.m2acdec
    0 references
    Q11125
    Defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; z ) Gauss-hypergeometric-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝑧 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
    xml-id
    C16.S2.m1abdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11691
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C16.S1.XMD2.m1fdec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1ddec
    0 references
    Q11693
    Defining formula
    a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
    xml-id
    C16.S1.XMD4.m1edec
    0 references
    Q11694
    Defining formula
    b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
    xml-id
    C16.S1.XMD5.m1edec
    0 references
    formal infinite series (locally)
    Defining formula
    E p , q ⁢ ( z ) subscript 𝐸 𝑝 𝑞 𝑧 {\displaystyle{\displaystyle E_{p,q}(z)}}
    xml-id
    C16.S11.XMD1.m1bdec
    0 references
    formal infinite series (locally)
    Defining formula
    H p , q ⁢ ( z ) subscript 𝐻 𝑝 𝑞 𝑧 {\displaystyle{\displaystyle H_{p,q}(z)}}
    xml-id
    C16.S11.XMD2.m1cdec
    0 references
    DLMF:16.11.E5
    Defining formula
    e k , m subscript 𝑒 𝑘 𝑚 {\displaystyle{\displaystyle e_{k,m}}}
    xml-id
    C16.S11.XMD5.m1bdec
    0 references
     
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