DLMF:15.2#Ex1 (Q4980)

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DLMF:15.2#Ex1
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    Statements

    Defining formula
    lim c → - n ⁡ F ⁡ ( a , b ; c ; z ) Γ ⁡ ( c ) = 𝐅 ⁡ ( a , b ; - n ; z ) = ( a ) n + 1 ⁢ ( b ) n + 1 ( n + 1 ) ! ⁢ z n + 1 ⁢ F ⁡ ( a + n + 1 , b + n + 1 ; n + 2 ; z ) , subscript → 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Euler-Gamma 𝑐 scaled-hypergeometric-bold-F 𝑎 𝑏 𝑛 𝑧 Pochhammer 𝑎 𝑛 1 Pochhammer 𝑏 𝑛 1 𝑛 1 superscript 𝑧 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑛 1 𝑏 𝑛 1 𝑛 2 𝑧 {\displaystyle{\displaystyle\lim_{c\to-n}\frac{F\left(a,b;c;z\right)}{\Gamma% \left(c\right)}=\mathbf{F}\left(a,b;-n;z\right)=\frac{{\left(a\right)_{n+1}}{% \left(b\right)_{n+1}}}{(n+1)!}z^{n+1}F\left(a+n+1,b+n+1;n+2;z\right),}}
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    DLMF id
    DLMF:15.2#Ex1
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    constraint
    n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots}}
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