DLMF:15.2.E1 (Q4977): Difference between revisions

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Property / Symbols defined
 
Property / Symbols defined: $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function / rank
 
Normal rank

Revision as of 15:18, 30 December 2019

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DLMF:15.2.E1
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    F ⁡ ( a , b ; c ; z ) = ∑ s = 0 ∞ ( a ) s ⁢ ( b ) s ( c ) s ⁢ s ! ⁢ z s = 1 + a ⁢ b c ⁢ z + a ⁢ ( a + 1 ) ⁢ b ⁢ ( b + 1 ) c ⁢ ( c + 1 ) ⁢ 2 ! ⁢ z 2 + ⋯ = Γ ⁡ ( c ) Γ ⁡ ( a ) ⁢ Γ ⁡ ( b ) ⁢ ∑ s = 0 ∞ Γ ⁡ ( a + s ) ⁢ Γ ⁡ ( b + s ) Γ ⁡ ( c + s ) ⁢ s ! ⁢ z s , Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript subscript 𝑠 0 Pochhammer 𝑎 𝑠 Pochhammer 𝑏 𝑠 Pochhammer 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 1 𝑎 𝑏 𝑐 𝑧 𝑎 𝑎 1 𝑏 𝑏 1 𝑐 𝑐 1 2 superscript 𝑧 2 ⋯ Euler-Gamma 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑏 superscript subscript 𝑠 0 Euler-Gamma 𝑎 𝑠 Euler-Gamma 𝑏 𝑠 Euler-Gamma 𝑐 𝑠 𝑠 superscript 𝑧 𝑠 {\displaystyle{\displaystyle F\left(a,b;c;z\right)=\sum_{s=0}^{\infty}\frac{{% \left(a\right)_{s}}{\left(b\right)_{s}}}{{\left(c\right)_{s}}s!}z^{s}=1+\frac{% ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}z^{2}+\cdots=\frac{\Gamma\left(c\right)}{% \Gamma\left(a\right)\Gamma\left(b\right)}\sum_{s=0}^{\infty}\frac{\Gamma\left(% a+s\right)\Gamma\left(b+s\right)}{\Gamma\left(c+s\right)s!}z^{s},}}
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    DLMF id
    DLMF:15.2.E1
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