DLMF:18.26.E19 (Q5949): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11581 / rank
 
Normal rank
Property / Symbols used: Q11581 / qualifier
 
test:

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)}}

\genhyperF{2}{1}@{\NVar{a},\NVar{b}}{\NVar{c}}{\NVar{z}}
Property / Symbols used: Q11581 / qualifier
 
xml-id: C16.S2.m5aadec
Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
test:

( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}

\Pochhammersym{\NVar{a}}{\NVar{n}}
Property / Symbols used: Q10759 / qualifier
 
xml-id: C5.S2.SS3.m1addec
Property / Symbols used
 
Property / Symbols used: Q11784 / rank
 
Normal rank
Property / Symbols used: Q11784 / qualifier
 
test:

S n ⁡ ( x ; a , b , c ) continuous-dual-Hahn-S 𝑛 𝑥 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c}\right)}}

\contdualHahnpolyS{\NVar{n}}@{\NVar{x}}{\NVar{a}}{\NVar{b}}{\NVar{c}}
Property / Symbols used: Q11784 / qualifier
 
xml-id: C18.S25.T1.t1.r3.m2addec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5addec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11743 / rank
 
Normal rank
Property / Symbols used: Q11743 / qualifier
 
test:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11743 / qualifier
 
xml-id: C18.S1.XMD1.m1idec
Property / Symbols used
 
Property / Symbols used: Q11750 / rank
 
Normal rank
Property / Symbols used: Q11750 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11750 / qualifier
 
xml-id: C18.S1.XMD2.m1adec
Property / Symbols used
 
Property / Symbols used: Q11726 / rank
 
Normal rank
Property / Symbols used: Q11726 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11726 / qualifier
 
xml-id: C18.S1.XMD6.m1qdec

Latest revision as of 00:46, 2 January 2020

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Language Label Description Also known as
English
DLMF:18.26.E19
No description defined

    Statements

    test
    ( 1 - z ) - c + i ⁢ y ⁢ F 1 2 ⁡ ( a + i ⁢ y , b + i ⁢ y a + b ; z ) = ∑ n = 0 ∞ S n ⁡ ( y 2 ; a , b , c ) ( a + b ) n ⁢ n ! ⁢ z n , superscript 1 𝑧 𝑐 𝑖 𝑦 Gauss-hypergeometric-F-as-2F1 𝑎 𝑖 𝑦 𝑏 𝑖 𝑦 𝑎 𝑏 𝑧 superscript subscript 𝑛 0 continuous-dual-Hahn-S 𝑛 superscript 𝑦 2 𝑎 𝑏 𝑐 Pochhammer 𝑎 𝑏 𝑛 𝑛 superscript 𝑧 𝑛 {\displaystyle{\displaystyle(1-z)^{-c+iy}{{}_{2}F_{1}}\left({a+iy,b+iy\atop a+% b};z\right)=\sum_{n=0}^{\infty}\frac{S_{n}\left(y^{2};a,b,c\right)}{{\left(a+b% \right)_{n}}n!}z^{n},}}
    0 references
    DLMF id
    DLMF:18.26.E19
    0 references
    constraint
    | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}}
    0 references
    Symbols used
    Q11581
    test
    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
    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1addec
    0 references
    Q11784
    test
    S n ⁡ ( x ; a , b , c ) continuous-dual-Hahn-S 𝑛 𝑥 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},% \NVar{c}\right)}}
    xml-id
    C18.S25.T1.t1.r3.m2addec
    0 references
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5addec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2afdec
    0 references
    Q11743
    test
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C18.S1.XMD1.m1idec
    0 references
    Q11750
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C18.S1.XMD2.m1adec
    0 references
    Q11726
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1qdec
    0 references
     
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