DLMF:16.12.E3 (Q5248): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11692 / rank
 
Normal rank
Property / Symbols used: Q11692 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11692 / qualifier
 
xml-id: C16.S1.XMD3.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11693 / rank
 
Normal rank
Property / Symbols used: Q11693 / qualifier
 
Defining formula:

a , a 1 , … , a p 𝑎 subscript 𝑎 1 … subscript 𝑎 𝑝 {\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}

a,a_{1},\ldots,a_{p}
Property / Symbols used: Q11693 / qualifier
 
xml-id: C16.S1.XMD4.m1bdec
Property / Symbols used
 
Property / Symbols used: Q11694 / rank
 
Normal rank
Property / Symbols used: Q11694 / qualifier
 
Defining formula:

b , b 1 , … , b q 𝑏 subscript 𝑏 1 … subscript 𝑏 𝑞 {\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}

b,b_{1},\ldots,b_{q}
Property / Symbols used: Q11694 / qualifier
 
xml-id: C16.S1.XMD5.m1bdec

Latest revision as of 14:40, 2 January 2020

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English
DLMF:16.12.E3
No description defined

    Statements

    Defining formula
    ( F 1 2 ⁡ ( a , b c ; z ) ) 2 = ∑ k = 0 ∞ ( 2 ⁢ a ) k ⁢ ( 2 ⁢ b ) k ⁢ ( c - 1 2 ) k ( c ) k ⁢ ( 2 ⁢ c - 1 ) k ⁢ k ! ⁢ F 3 4 ⁡ ( - 1 2 ⁢ k , 1 2 ⁢ ( 1 - k ) , a + b - c + 1 2 , 1 2 a + 1 2 , b + 1 2 , 3 2 - k - c ; 1 ) ⁢ z k , superscript Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 2 superscript subscript 𝑘 0 Pochhammer 2 𝑎 𝑘 Pochhammer 2 𝑏 𝑘 Pochhammer 𝑐 1 2 𝑘 Pochhammer 𝑐 𝑘 Pochhammer 2 𝑐 1 𝑘 𝑘 Gauss-hypergeometric-pFq 4 3 1 2 𝑘 1 2 1 𝑘 𝑎 𝑏 𝑐 1 2 1 2 𝑎 1 2 𝑏 1 2 3 2 𝑘 𝑐 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)% \right)^{2}=\sum_{k=0}^{\infty}\frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}% }{\left(c-\frac{1}{2}\right)_{k}}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}% k!}{{}_{4}F_{3}}\left({-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{% 1}{2}\atop a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},}}
    0 references
    DLMF id
    DLMF:16.12.E3
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    Symbols used
    Q11581
    Defining formula
    F 1 2 ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};% \NVar{z}\right)}}
    xml-id
    C16.S2.m5aadec
    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
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11692
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C16.S1.XMD3.m1bdec
    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.m1bdec
    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.m1bdec
    0 references
     
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