Formula:5764

Digital Library of Mathematical Functions ID 15.8.E10

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

Constraint(s)

${\displaystyle{\displaystyle|z-1|<1,|\operatorname{ph}\left(1-z\right)|<\pi}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: Olver’s hypergeometric function
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
• ${\displaystyle{\displaystyle b}}$: real or complex parameter
• ${\displaystyle{\displaystyle k}}$: integer
• ${\displaystyle{\displaystyle m}}$: integer
 

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