DLMF:16.13.E4 (Q5252): Difference between revisions

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Property / Defining formulaProperty / Defining formula

F 4 ⁡ ( α , β ; γ , γ ′ ; x , y ) = ∑ m , n = 0 ∞ ( α ) m + n ⁢ ( β ) m + n ( γ ) m ⁢ ( γ ′ ) n ⁢ m ! ⁢ n ! ⁢ x m ⁢ y n , Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ 𝑥 𝑦 superscript subscript 𝑚 𝑛 0 Pochhammer 𝛼 𝑚 𝑛 Pochhammer 𝛽 𝑚 𝑛 Pochhammer 𝛾 𝑚 Pochhammer superscript 𝛾 ′ 𝑛 𝑚 𝑛 superscript 𝑥 𝑚 superscript 𝑦 𝑛 {\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x% ,y\right)=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta% \right)_{m+n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n% !}x^{m}y^{n},}}

\AppellF{4}@{\alpha}{\beta}{\gamma}{\gamma^{\prime}}{x}{y}=\sum_{m,n=0}^{\infty}\frac{\Pochhammersym{\alpha}{m+n}\Pochhammersym{\beta}{m+n}}{\Pochhammersym{\gamma}{m}\Pochhammersym{\gamma^{\prime}}{n}m!n!}x^{m}y^{n},

F 4 ⁡ ( α , β ; γ , γ ′ ; x , y ) = ∑ m , n = 0 ∞ ( α ) m + n ⁢ ( β ) m + n ( γ ) m ⁢ ( γ ′ ) n ⁢ m ! ⁢ n ! ⁢ x m ⁢ y n Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ 𝑥 𝑦 superscript subscript 𝑚 𝑛 0 Pochhammer 𝛼 𝑚 𝑛 Pochhammer 𝛽 𝑚 𝑛 Pochhammer 𝛾 𝑚 Pochhammer superscript 𝛾 ′ 𝑛 𝑚 𝑛 superscript 𝑥 𝑚 superscript 𝑦 𝑛 {\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x% ,y\right)=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta% \right)_{m+n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n% !}x^{m}y^{n}}}

\AppellF{4}@{\alpha}{\beta}{\gamma}{\gamma^{\prime}}{x}{y}=\sum_{m,n=0}^{\infty}\frac{\Pochhammersym{\alpha}{m+n}\Pochhammersym{\beta}{m+n}}{\Pochhammersym{\gamma}{m}\Pochhammersym{\gamma^{\prime}}{n}m!n!}x^{m}y^{n}
Property / Symbols defined
 
Property / Symbols defined: fourth Appell function / rank
 
Normal rank
Property / constraint
 

| x | + | y | < 1 𝑥 𝑦 1 {\displaystyle{\displaystyle\sqrt{|x|}+\sqrt{|y|}<1}}

\sqrt{|x|}+\sqrt{|y|}<1
Property / constraint: | x | + | y | < 1 𝑥 𝑦 1 {\displaystyle{\displaystyle\sqrt{|x|}+\sqrt{|y|}<1}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10759 / rank
 
Normal rank
Property / Symbols used: Q10759 / qualifier
 
Defining formula:

( 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.m1acdec
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
Defining formula:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5acdec

Latest revision as of 14:40, 2 January 2020

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DLMF:16.13.E4
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    Statements

    Defining formula
    F 4 ⁡ ( α , β ; γ , γ ′ ; x , y ) = ∑ m , n = 0 ∞ ( α ) m + n ⁢ ( β ) m + n ( γ ) m ⁢ ( γ ′ ) n ⁢ m ! ⁢ n ! ⁢ x m ⁢ y n Appell-F-4 𝛼 𝛽 𝛾 superscript 𝛾 ′ 𝑥 𝑦 superscript subscript 𝑚 𝑛 0 Pochhammer 𝛼 𝑚 𝑛 Pochhammer 𝛽 𝑚 𝑛 Pochhammer 𝛾 𝑚 Pochhammer superscript 𝛾 ′ 𝑛 𝑚 𝑛 superscript 𝑥 𝑚 superscript 𝑦 𝑛 {\displaystyle{\displaystyle{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x% ,y\right)=\sum_{m,n=0}^{\infty}\frac{{\left(\alpha\right)_{m+n}}{\left(\beta% \right)_{m+n}}}{{\left(\gamma\right)_{m}}{\left(\gamma^{\prime}\right)_{n}}m!n% !}x^{m}y^{n}}}
    0 references
    DLMF id
    DLMF:16.13.E4
    0 references
    constraint
    | x | + | y | < 1 𝑥 𝑦 1 {\displaystyle{\displaystyle\sqrt{|x|}+\sqrt{|y|}<1}}
    0 references
    Symbols used
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5acdec
    0 references
     
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