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

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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}

Revision as of 15:23, 30 December 2019

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DLMF:16.13.E4
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    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}}}
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    DLMF:16.13.E4
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