DLMF:15.8.E10 (Q5067)

From DRMF

Jump to navigation Jump to search

No description defined

Language	Label	Description	Also known as
English	DLMF:15.8.E10	No description defined

Statements

Defining formula

{\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),}}

0 references

DLMF:15.8.E10

0 references

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

0 references

Defining formula

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

C5.S2.E1.m2afdec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

{\displaystyle{\displaystyle\pi}}

C3.S12.E1.m2agdec

0 references

psi (or digamma) function

Defining formula

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

C5.S2.E2.m2abdec

0 references

Defining formula

{\displaystyle{\displaystyle!}}

introduction.Sx4.p1.t1.r15.m5abdec

0 references

principal branch of logarithm function

Defining formula

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

C4.S2.E2.m2abdec

0 references

Defining formula

{\displaystyle{\displaystyle\operatorname{ph}}}

C1.S9.E7.m1agdec

0 references

$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function

Defining formula

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

C15.S2.E2.m2agdec

0 references

Defining formula

{\displaystyle{\displaystyle z}}

C15.S1.XMD3.m1idec

0 references

Defining formula

{\displaystyle{\displaystyle a}}

C15.S1.XMD4.m1gdec

0 references

Defining formula

{\displaystyle{\displaystyle b}}

C15.S1.XMD5.m1gdec

0 references

Defining formula

{\displaystyle{\displaystyle k}}

C15.S1.XMD7.m1bdec

0 references

Defining formula

{\displaystyle{\displaystyle m}}

C15.S1.XMD9.m1ddec

0 references

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q5067&oldid=42935"