15 Hypergeometric Function Properties 15.3 Graphics 15.5 Derivatives and Contiguous Functions

§15.4 Special Cases

Permalink:: http://dlmf.nist.gov/15.4

§15.4(i) Elementary Functions
§15.4(ii) Argument Unity
§15.4(iii) Other Arguments

§15.4(i) Elementary Functions

Notes:: For (15.4.1)–(15.4.6) see Temme (1996b, §5.1). For (15.4.7) use (15.8.27), (15.4.6), and (5.5.5). For (15.4.9) use (15.8.28), (15.4.6), and (5.5.5). For (15.4.11) use (15.8.1) and (15.4.7). For (15.4.13) use (15.8.1) and (15.4.11). For (15.4.15) use (15.8.1) and (15.4.9). For (15.4.17) and (15.4.18) use (15.8.15) and (15.4.6). For (15.4.19) use (15.5.11) and (15.4.6).
A&S Ref:: 15.1
Keywords:: elementary functions, hypergeometric function
Permalink:: http://dlmf.nist.gov/15.4.i

The following results hold for principal branches when , and by analytic continuation elsewhere. Exceptions are (15.4.8) and (15.4.10), that hold for $|z|<\ifrac{\pi}{4}$ , and (15.4.12), (15.4.14), and (15.4.16), that hold for $|z|<\ifrac{\pi}{2}$ .

15.4.1 $\mathop{F\/}\nolimits\!\left(1,1;2;z\right)=-z^{{-1}}\mathop{\ln\/}\nolimits\!% \left(1-z\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E1
Encodings:: TeX, pMML, png

15.4.2 $\mathop{F\/}\nolimits\!\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)=\frac{1}% {2z}\mathop{\ln\/}\nolimits\!\left(\frac{1+z}{1-z}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E2
Encodings:: TeX, pMML, png

15.4.3 $\mathop{F\/}\nolimits\!\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)=z^{{-1}% }\mathop{\mathrm{arctan}\/}\nolimits z,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function and : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E3
Encodings:: TeX, pMML, png

15.4.4 $\mathop{F\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}% \right)=z^{{-1}}\mathop{\mathrm{arcsin}\/}\nolimits z,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function and : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E4
Encodings:: TeX, pMML, png

15.4.5 $\mathop{F\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}% \right)=z^{{-1}}\mathop{\ln\/}\nolimits\!\left(z+\sqrt{1+z^{2}}\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and : complex variable
Permalink:: http://dlmf.nist.gov/15.4.E5
Encodings:: TeX, pMML, png

15.4.6 $\mathop{F\/}\nolimits\!\left(a,b;b;z\right)=(1-z)^{{-a}};$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §10.22(iv), §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E6
Encodings:: TeX, pMML, png

compare §15.2(ii).

15.4.7 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac% {1}{2}\left((1+z)^{{-2a}}+(1-z)^{{-2a}}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E7
Encodings:: TeX, pMML, png

15.4.8 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\mathop{\tan\/}% \nolimits^{{2}}}z\right)=(\mathop{\cos\/}\nolimits z)^{{2a}}\mathop{\cos\/}% \nolimits\!\left(2az\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\tan\/}\nolimits z$ : tangent function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E8
Encodings:: TeX, pMML, png

15.4.9 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{% 1}{(2-4a)z}\left((1+z)^{{1-2a}}-(1-z)^{{1-2a}}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E9
Encodings:: TeX, pMML, png

15.4.10 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\mathop{\tan\/}% \nolimits^{{2}}}z\right)=(\mathop{\cos\/}\nolimits z)^{{2a}}\frac{\mathop{\sin% \/}\nolimits\!\left((1-2a)z\right)}{(1-2a)\mathop{\sin\/}\nolimits z}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E10
Encodings:: TeX, pMML, png

15.4.11 $\mathop{F\/}\nolimits\!\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left% (\left(\sqrt{1+z^{2}}+z\right)^{{2a}}+\left(\sqrt{1+z^{2}}-z\right)^{{2a}}% \right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E11
Encodings:: TeX, pMML, png

15.4.12 $\mathop{F\/}\nolimits\!\left(-a,a;\tfrac{1}{2};{\mathop{\sin\/}\nolimits^{{2}}% }z\right)=\mathop{\cos\/}\nolimits\!\left(2az\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E12
Encodings:: TeX, pMML, png

15.4.13 $\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{% 1+z^{2}}}\left(\left(\sqrt{1+z^{2}}+z\right)^{{2a-1}}+\left(\sqrt{1+z^{2}}-z% \right)^{{2a-1}}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E13
Encodings:: TeX, pMML, png

15.4.14 $\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{1}{2};{\mathop{\sin\/}\nolimits^{{2}% }}z\right)=\frac{\mathop{\cos\/}\nolimits\!\left((2a-1)z\right)}{\mathop{\cos% \/}\nolimits z}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E14
Encodings:: TeX, pMML, png

15.4.15 $\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z% }\left(\left(\sqrt{1+z^{2}}+z\right)^{{1-2a}}-\left(\sqrt{1+z^{2}}-z\right)^{{% 1-2a}}\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E15
Encodings:: TeX, pMML, png

15.4.16 $\mathop{F\/}\nolimits\!\left(a,1-a;\tfrac{3}{2};{\mathop{\sin\/}\nolimits^{{2}% }}z\right)=\frac{\mathop{\sin\/}\nolimits\!\left((2a-1)z\right)}{(2a-1)\mathop% {\sin\/}\nolimits z}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E16
Encodings:: TeX, pMML, png

15.4.17 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}% +\tfrac{1}{2}\sqrt{1-z}\right)^{{-2a}},$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E17
Encodings:: TeX, pMML, png

15.4.18 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}% \left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z}\right)^{{1-2a}}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E18
Encodings:: TeX, pMML, png

15.4.19 $\mathop{F\/}\nolimits\!\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z% \right)(1-z)^{{-1-b}}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(i)
Permalink:: http://dlmf.nist.gov/15.4.E19
Encodings:: TeX, pMML, png

For an extensive list of elementary representations see Prudnikov et al. (1990, pp. 468–488).

§15.4(ii) Argument Unity

Notes:: See Andrews et al. (1999, §2.2). For (15.4.21) use (15.8.10). For (15.4.22) and (15.4.23) use (15.8.4), (5.5.3), and (5.5.1). For (15.4.25) see Dougall (1907) or Andrews et al. (1999, §2.8).
Keywords:: gamma function, hypergeometric function
Referenced by:: §15.2(i)
Permalink:: http://dlmf.nist.gov/15.4.ii

If $\realpart{(c-a-b)}>0$ , then

15.4.20 $\mathop{F\/}\nolimits\!\left(a,b;c;1\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(c\right)\mathop{\Gamma\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/% }\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real or complex parameter, : real or complex parameter and : real or complex parameter
A&S Ref:: 15.1.20
Referenced by:: §13.23(i), ¶ ‣ §15.4(ii), §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E20
Encodings:: TeX, pMML, png

If , then

15.4.21 $\lim_{{z\to 1-}}\frac{\mathop{F\/}\nolimits\!\left(a,b;a+b;z\right)}{-\mathop{% \ln\/}\nolimits\!\left(1-z\right)}=\frac{\mathop{\Gamma\/}\nolimits\!\left(a+b% \right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits% \!\left(b\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, : complex variable, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E21
Encodings:: TeX, pMML, png

If $\realpart{(c-a-b)}=0$ and $c\neq a+b$ , then

15.4.22 $\lim_{{z\to 1-}}(1-z)^{{a+b-c}}\left(\mathop{F\/}\nolimits\!\left(a,b;c;z% \right)-\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(c-a-b\right)}{\mathop{\Gamma\/}\nolimits\!\left(c-a\right)% \mathop{\Gamma\/}\nolimits\!\left(c-b\right)}\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(a+b-c\right)}{% \mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b% \right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E22
Encodings:: TeX, pMML, png

If $\realpart{(c-a-b)}<0$ , then

15.4.23 $\lim_{{z\to 1-}}\frac{\mathop{F\/}\nolimits\!\left(a,b;c;z\right)}{(1-z)^{{c-a% -b}}}=\frac{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}% \nolimits\!\left(a+b-c\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)% \mathop{\Gamma\/}\nolimits\!\left(b\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : complex variable, : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E23
Encodings:: TeX, pMML, png

¶ Chu–Vandermonde Identity

Keywords:: Chu–Vandermonde identity, hypergeometric function

15.4.24 $\mathop{F\/}\nolimits\!\left(-n,b;c;1\right)=\frac{\left(c-b\right)_{{n}}}{% \left(c\right)_{{n}}},$ $n=0,1,2,\dots$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, : integer, : real or complex parameter and : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.4.E24
Encodings:: TeX, pMML, png

¶ Dougall’s Bilateral Sum

Keywords:: Dougall’s bilateral sum

This is a generalization of (15.4.20). If are not integers and $\realpart{(c+d-a-b)}>1$ , then

15.4.25 $\sum_{{n=-\infty}}^{{\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+n\right% )\mathop{\Gamma\/}\nolimits\!\left(b+n\right)}{\mathop{\Gamma\/}\nolimits\!% \left(c+n\right)\mathop{\Gamma\/}\nolimits\!\left(d+n\right)}=\frac{\pi^{2}}{% \mathop{\sin\/}\nolimits\!\left(\pi a\right)\mathop{\sin\/}\nolimits\!\left(% \pi b\right)}\*\frac{\mathop{\Gamma\/}\nolimits\!\left(c+d-a-b-1\right)}{% \mathop{\Gamma\/}\nolimits\!\left(c-a\right)\mathop{\Gamma\/}\nolimits\!\left(% d-a\right)\mathop{\Gamma\/}\nolimits\!\left(c-b\right)\mathop{\Gamma\/}% \nolimits\!\left(d-b\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : real or complex parameter, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.E25
Encodings:: TeX, pMML, png

§15.4(iii) Other Arguments

Notes:: For (15.4.26) use (15.8.24) and (5.5.5). For (15.4.27) use (15.2.1) and (5.7.7). For (15.4.28) use (15.8.1), (15.4.26), and (5.5.5). For (15.4.29) use (15.8.1), (15.5.15), (15.4.26), (5.5.5), and (5.5.1). For (15.4.30) use (15.8.1) and (15.4.28). For (15.4.31) use (15.8.1), Erdélyi et al. (1953a, Eq. (2.11.41)), (15.4.20), and (5.5.5). For (15.4.32) use (15.8.30), (15.4.28), (5.5.5), and (5.5.6). (Note that Erdélyi et al. (1953a, Eq. (2.8.54)) contains an error.) For (15.4.33) let $z\to-\infty$ in (15.8.32) and use (5.5.5).
A&S Ref:: 15.1
Keywords:: gamma function, hypergeometric function, psi function
Permalink:: http://dlmf.nist.gov/15.4.iii

15.4.26 $\mathop{F\/}\nolimits\!\left(a,b;a-b+1;-1\right)=\frac{\mathop{\Gamma\/}% \nolimits\!\left(a-b+1\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+1% \right)}{\mathop{\Gamma\/}\nolimits\!\left(a+1\right)\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}a-b+1\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E26
Encodings:: TeX, pMML, png

15.4.27 $\mathop{F\/}\nolimits\!\left(1,a;a+1;-1\right)=\tfrac{1}{2}a\left(\mathop{\psi% \/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\mathop{\psi\/}\nolimits% \!\left(\tfrac{1}{2}a\right)\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E27
Encodings:: TeX, pMML, png

15.4.28 $\mathop{F\/}\nolimits\!\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};% \tfrac{1}{2}\right)=\sqrt{\pi}\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1% }{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}a+\tfrac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2% }b+\tfrac{1}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E28
Encodings:: TeX, pMML, png

15.4.29 $\mathop{F\/}\nolimits\!\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}% \right)=\frac{2\sqrt{\pi}}{a-b}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a% +\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}a\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b+\tfrac{1}{% 2}\right)}-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}a+\tfrac{1}{% 2}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}b\right)}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E29
Encodings:: TeX, pMML, png

15.4.30 $\mathop{F\/}\nolimits\!\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{{1-b}}\sqrt{% \pi}\mathop{\Gamma\/}\nolimits\!\left(b\right)}{\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}b-\tfrac{1}{2}a+\tfrac{1}{2}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : real or complex parameter and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E30
Encodings:: TeX, pMML, png

15.4.31 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}% \right)=\left(\frac{8}{9}\right)^{{-2a}}\frac{\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{4}{3}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}-2a% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}\right)\mathop{\Gamma\/% }\nolimits\!\left(\tfrac{4}{3}-2a\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E31
Encodings:: TeX, pMML, png

15.4.32 $\mathop{F\/}\nolimits\!\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;% \tfrac{1}{9}\right)=\sqrt{\pi}\left(\frac{3}{4}\right)^{a}\frac{\mathop{\Gamma% \/}\nolimits\!\left(\tfrac{5}{6}+\tfrac{2}{3}a\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{5}{6}+\tfrac{1}{3}a\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E32
Encodings:: TeX, pMML, png

15.4.33 $\mathop{F\/}\nolimits\!\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;e^{{\ifrac{i\pi% }{3}}}\right)=\sqrt{\pi}e^{{\ifrac{i\pi a}{2}}}\left(\frac{16}{27}\right)^{{(3% a+1)/6}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{5}{6}+a\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\frac{2}{3}+a\right)\mathop{\Gamma\/}\nolimits\!% \left(\frac{2}{3}\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, : base of exponential function and : real or complex parameter
Referenced by:: §15.4(iii)
Permalink:: http://dlmf.nist.gov/15.4.E33
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 15.3 Graphics 15.5 Derivatives and Contiguous Functions