DLMF:15.10.E9 (Q5125)

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Language	Label	Description	Also known as
English	DLMF:15.10.E9	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle F\left({-m,b\atop n};z\right)\ln z-\sum_{k=1}^{n-% 1}\frac{(n-1)!(k-1)!}{(n-k-1)!{\left(m+1\right)_{k}}{\left(1-b\right)_{k}}}(-z% )^{-k}+\sum_{k=0}^{m}\frac{{\left(-m\right)_{k}}{\left(b\right)_{k}}}{{\left(n% \right)_{k}}k!}z^{k}\left(\psi\left(1+m-k\right)+\psi\left(b+k\right)-\psi% \left(1+k\right)-\psi\left(n+k\right)\right)+(-1)^{m}m!\sum_{k=m+1}^{\infty}% \frac{(k-1-m)!{\left(b\right)_{k}}}{{\left(n\right)_{k}}k!}z^{k},}}

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DLMF:15.10.E9

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{\displaystyle{\displaystyle b\neq n-1,n-2,\dots,0,-1,-2,\dots}}

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$$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function

Defining formula

{\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}

C15.S2.E1.m2addec

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Defining formula

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

C5.S2.SS3.m1aadec

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psi (or digamma) function

Defining formula

{\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

C5.S2.E2.m2aadec

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Defining formula

{\displaystyle{\displaystyle!}}

introduction.Sx4.p1.t1.r15.m5aadec

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principal branch of logarithm function

Defining formula

{\displaystyle{\displaystyle\ln\NVar{z}}}

C4.S2.E2.m2aadec

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Defining formula

{\displaystyle{\displaystyle n}}

C15.S1.XMD10.m1adec

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Defining formula

{\displaystyle{\displaystyle z}}

C15.S1.XMD3.m1hdec

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Defining formula

{\displaystyle{\displaystyle a}}

C15.S1.XMD4.m1hdec

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Defining formula

{\displaystyle{\displaystyle b}}

C15.S1.XMD5.m1hdec

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Defining formula

{\displaystyle{\displaystyle k}}

C15.S1.XMD7.m1adec

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Defining formula

{\displaystyle{\displaystyle m}}

C15.S1.XMD9.m1adec

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