15 Hypergeometric Function Properties 15.3 Graphics 15.5 Derivatives and Contiguous Functions

§15.4 Special Cases

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Referenced by:: §15.2(ii)
Permalink:: http://dlmf.nist.gov/15.4
See also:: Annotations for Ch.15

§15.4(i) Elementary Functions

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Keywords:: elementary functions, hypergeometric function, relations to other functions, special cases
A&S Ref:: 15.1
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).
Referenced by:: Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.i
Clarification (effective with 1.0.16):: Information was added after (15.4.6) and (15.4.19) specifying that special care is required under certain circumstances.
Suggested 2017-01-01 by Louis Klauder
See also:: Annotations for §15.4 and Ch.15

The following results hold for principal branches when $|z|<1$ , 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	$\displaystyle F\left(1,1;2;z\right)$	$\displaystyle=-z^{-1}\ln\left(1-z\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E1 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15
15.4.2	$\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};z^{2}\right)$	$\displaystyle=\frac{1}{2z}\ln\left(\frac{1+z}{1-z}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable Permalink: http://dlmf.nist.gov/15.4.E2 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15
15.4.3	$\displaystyle F\left(\tfrac{1}{2},1;\tfrac{3}{2};-z^{2}\right)$	$\displaystyle=z^{-1}\operatorname{arctan}z,$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\operatorname{arctan}\NVar{z}$ : arctangent function and $z$ : complex variable Permalink: http://dlmf.nist.gov/15.4.E3 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15
15.4.4	$\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};z^{2}\right)$	$\displaystyle=z^{-1}\operatorname{arcsin}z,$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $z$ : complex variable Permalink: http://dlmf.nist.gov/15.4.E4 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15
15.4.5	$\displaystyle F\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}\right)$	$\displaystyle=z^{-1}\ln\left(z+\sqrt{1+z^{2}}\right).$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable Permalink: http://dlmf.nist.gov/15.4.E5 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.6	$\displaystyle F\left(a,b;a;z\right)$	$\displaystyle=(1-z)^{-b},$
15.4.6	$\displaystyle F\left(a,b;b;z\right)$	$\displaystyle=(1-z)^{-a},$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §10.22(iv), §15.2(ii), §15.4(i), (18.12.3_5), Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii) Permalink: http://dlmf.nist.gov/15.4.E6 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.16): This equation was expanded by adding the formula $F\left(a,b;a;z\right)=(1-z)^{-b}$ . See also: Annotations for §15.4(i), §15.4 and Ch.15

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when the third parameter is a nonpositive integer. See the final paragraph in §15.2(ii).

15.4.7		$F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};z^{2}\right)=\tfrac{1}{2}\left((1+z)^{-2a% }+(1-z)^{-2a}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E7 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.8		$F\left(a,\tfrac{1}{2}+a;\tfrac{1}{2};-{\tan}^{2}z\right)=(\cos z)^{2a}\cos% \left(2az\right).$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\cos\NVar{z}$ : cosine function, $\tan\NVar{z}$ : tangent function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E8 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.9		$F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};z^{2}\right)=\frac{1}{(2-4a)z}\left((1+z)% ^{1-2a}-(1-z)^{1-2a}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E9 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.10		$F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2};-{\tan}^{2}z\right)=(\cos z)^{2a}\frac{% \sin\left((1-2a)z\right)}{(1-2a)\sin z}.$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E10 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.11		$F\left(-a,a;\tfrac{1}{2};-z^{2}\right)=\tfrac{1}{2}\left(\left(\sqrt{1+z^{2}}+% z\right)^{2a}+\left(\sqrt{1+z^{2}}-z\right)^{2a}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E11 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.12		$F\left(-a,a;\tfrac{1}{2};{\sin}^{2}z\right)=\cos\left(2az\right).$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E12 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.13		$F\left(a,1-a;\tfrac{1}{2};-z^{2}\right)=\frac{1}{2\sqrt{1+z^{2}}}\left(\left(% \sqrt{1+z^{2}}+z\right)^{2a-1}+\left(\sqrt{1+z^{2}}-z\right)^{2a-1}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E13 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.14		$F\left(a,1-a;\tfrac{1}{2};{\sin}^{2}z\right)=\frac{\cos\left((2a-1)z\right)}{% \cos z}.$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E14 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.15		$F\left(a,1-a;\tfrac{3}{2};-z^{2}\right)=\frac{1}{(2-4a)z}\left(\left(\sqrt{1+z% ^{2}}+z\right)^{1-2a}-\left(\sqrt{1+z^{2}}-z\right)^{1-2a}\right),$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E15 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.16		$F\left(a,1-a;\tfrac{3}{2};{\sin}^{2}z\right)=\frac{\sin\left((2a-1)z\right)}{(% 2a-1)\sin z}.$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E16 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.17		$F\left(a,\tfrac{1}{2}+a;1+2a;z\right)=\left(\tfrac{1}{2}+\tfrac{1}{2}\sqrt{1-z% }\right)^{-2a},$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §15.4(i), §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E17 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.18		$F\left(a,\tfrac{1}{2}+a;2a;z\right)=\frac{1}{\sqrt{1-z}}\left(\tfrac{1}{2}+% \tfrac{1}{2}\sqrt{1-z}\right)^{1-2a},$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable and $a$ : real or complex parameter Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §15.4(i), §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E18 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

15.4.19		$F\left(a+1,b;a;z\right)=\left(1-(1-(\ifrac{b}{a}))z\right)(1-z)^{-1-b}.$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §15.4(i), §15.4(i) Permalink: http://dlmf.nist.gov/15.4.E19 Encodings: TeX, pMML, png See also: Annotations for §15.4(i), §15.4 and Ch.15

In (15.4.17), (15.4.18) and (15.4.19) when the third entry is a nonpositive integer one has to use the limit interpretation (15.2.6), rather than (15.2.5). Compare the final paragraph in §15.2(ii).

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

§15.4(ii) Argument Unity

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Keywords:: argument $\pm 1$ , gamma function, hypergeometric function, relations to hypergeometric function, relations to other functions, special cases
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).
Referenced by:: §15.2(i), Erratum (V1.0.16) for Subsections 15.4(i), 15.4(ii)
Permalink:: http://dlmf.nist.gov/15.4.ii
See also:: Annotations for §15.4 and Ch.15

If $\Re\left(c-a-b\right)>0$ , then

15.4.20		$F\left(a,b;c;1\right)=\frac{\Gamma\left(c\right)\Gamma\left(c-a-b\right)}{% \Gamma\left(c-a\right)\Gamma\left(c-b\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : 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 See also: Annotations for §15.4(ii), §15.4 and Ch.15

If $c=a+b$ , then

15.4.21		$\lim_{z\to 1-}\frac{F\left(a,b;a+b;z\right)}{-\ln\left(1-z\right)}=\frac{% \Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\ln\NVar{z}$ : principal branch of logarithm function, $z$ : complex variable, $a$ : real or complex parameter and $b$ : real or complex parameter Referenced by: §15.4(ii) Permalink: http://dlmf.nist.gov/15.4.E21 Encodings: TeX, pMML, png See also: Annotations for §15.4(ii), §15.4 and Ch.15

If $\Re\left(c-a-b\right)=0$ and $c\neq a+b$ , then

15.4.22		$\lim_{z\to 1-}(1-z)^{a+b-c}\left(F\left(a,b;c;z\right)-\frac{\Gamma\left(c% \right)\Gamma\left(c-a-b\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b\right)}% \right)=\frac{\Gamma\left(c\right)\Gamma\left(a+b-c\right)}{\Gamma\left(a% \right)\Gamma\left(b\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.4(ii) Permalink: http://dlmf.nist.gov/15.4.E22 Encodings: TeX, pMML, png See also: Annotations for §15.4(ii), §15.4 and Ch.15

If $\Re\left(c-a-b\right)<0$ , then

15.4.23		$\lim_{z\to 1-}\frac{F\left(a,b;c;z\right)}{(1-z)^{c-a-b}}=\frac{\Gamma\left(c% \right)\Gamma\left(a+b-c\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.4(ii) Permalink: http://dlmf.nist.gov/15.4.E23 Encodings: TeX, pMML, png See also: Annotations for §15.4(ii), §15.4 and Ch.15

Chu–Vandermonde Identity

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Keywords:: Chu–Vandermonde identity, hypergeometric function
See also:: Annotations for §15.4(ii), §15.4 and Ch.15

15.4.24		$F\left(-n,b;c;1\right)=\frac{{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}},$
$n=0,1,2,\dots$ .
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : integer, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.4.E24 Encodings: TeX, pMML, png See also: Annotations for §15.4(ii), §15.4(ii), §15.4 and Ch.15

Dougall’s Bilateral Sum

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Keywords:: Dougall’s bilateral sum
See also:: Annotations for §15.4(ii), §15.4 and Ch.15

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

15.4.25		$\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n\right)\Gamma\left(b+n\right)}{% \Gamma\left(c+n\right)\Gamma\left(d+n\right)}=\frac{\pi^{2}}{\sin\left(\pi a% \right)\sin\left(\pi b\right)}\*\frac{\Gamma\left(c+d-a-b-1\right)}{\Gamma% \left(c-a\right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b% \right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function, $n$ : integer, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.4(ii) Permalink: http://dlmf.nist.gov/15.4.E25 Encodings: TeX, pMML, png See also: Annotations for §15.4(ii), §15.4(ii), §15.4 and Ch.15

§15.4(iii) Other Arguments

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Keywords:: argument a fraction, argument $\pm 1$ , arguments ${\mathrm{e}}^{\pm\mathrm{i}\pi/3}$ , gamma function, hypergeometric function, psi function, relation to hypergeometric function, relations to hypergeometric function, relations to other functions, special cases
A&S Ref:: 15.1
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).
Permalink:: http://dlmf.nist.gov/15.4.iii
Addition (effective with 1.1.2):: Equation (15.4.34) and some explanatory text immediately after it were added.
Clarification (effective with 1.0.16):: Information was added after (15.4.33) specifying special care is required under certain circumstances.
See also:: Annotations for §15.4 and Ch.15

15.4.26		$F\left(a,b;a-b+1;-1\right)=\frac{\Gamma\left(a-b+1\right)\Gamma\left(\tfrac{1}% {2}a+1\right)}{\Gamma\left(a+1\right)\Gamma\left(\tfrac{1}{2}a-b+1\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $a$ : real or complex parameter and $b$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E26 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.27		$F\left(1,a;a+1;-1\right)=2^{-a}F\left(a,a;a+1;\tfrac{1}{2}\right)=\tfrac{1}{2}% a\left(\psi\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)-\psi\left(\tfrac{1}{2}a% \right)\right).$
ⓘ Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function and $a$ : real or complex parameter Proof sketch: Intermediate equality follows from (15.8.1). Referenced by: §15.4(iii), Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/15.4.E27 Encodings: TeX, pMML, png Addition (effective with 1.1.2): Intermediate equality was added. See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.28		$F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2};\tfrac{1}{2}\right)=\sqrt{% \pi}\frac{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+\tfrac{1}{2}\right)}{\Gamma% \left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}% \right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a$ : real or complex parameter and $b$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E28 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.29		$F\left(a,b;\tfrac{1}{2}a+\tfrac{1}{2}b+1;\tfrac{1}{2}\right)=\frac{2\sqrt{\pi}% }{a-b}\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b+1\right)\*\left(\frac{1}{\Gamma% \left(\tfrac{1}{2}a\right)\Gamma\left(\tfrac{1}{2}b+\tfrac{1}{2}\right)}-\frac% {1}{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}b% \right)}\right).$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a$ : real or complex parameter and $b$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E29 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.30		$F\left(a,1-a;b;\tfrac{1}{2}\right)=\frac{2^{1-b}\sqrt{\pi}\Gamma\left(b\right)% }{\Gamma\left(\tfrac{1}{2}a+\tfrac{1}{2}b\right)\Gamma\left(\tfrac{1}{2}b-% \tfrac{1}{2}a+\tfrac{1}{2}\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $a$ : real or complex parameter and $b$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E30 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.31		$F\left(a,\tfrac{1}{2}+a;\tfrac{3}{2}-2a;-\tfrac{1}{3}\right)=\left(\frac{8}{9}% \right)^{-2a}\frac{\Gamma\left(\tfrac{4}{3}\right)\Gamma\left(\tfrac{3}{2}-2a% \right)}{\Gamma\left(\tfrac{3}{2}\right)\Gamma\left(\tfrac{4}{3}-2a\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function and $a$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E31 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.32		$F\left(a,\tfrac{1}{2}+a;\tfrac{5}{6}+\tfrac{2}{3}a;\tfrac{1}{9}\right)=\sqrt{% \pi}\left(\frac{3}{4}\right)^{a}\frac{\Gamma\left(\tfrac{5}{6}+\tfrac{2}{3}a% \right)}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{3}a\right)\Gamma\left(\tfrac{5}{6}% +\tfrac{1}{3}a\right)}.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter and $a$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E32 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.33		$F\left(3a,\tfrac{1}{3}+a;\tfrac{2}{3}+2a;{\mathrm{e}}^{\ifrac{\mathrm{i}\pi}{3% }}\right)=\sqrt{\pi}{\mathrm{e}}^{\ifrac{\mathrm{i}\pi a}{2}}\left(\frac{16}{2% 7}\right)^{(3a+1)/6}\frac{\Gamma\left(\frac{5}{6}+a\right)}{\Gamma\left(\frac{% 2}{3}+a\right)\Gamma\left(\frac{2}{3}\right)},$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $a$ : real or complex parameter Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §15.4(iii), §15.4(iii) Permalink: http://dlmf.nist.gov/15.4.E33 Encodings: TeX, pMML, png See also: Annotations for §15.4(iii), §15.4 and Ch.15

15.4.34		$F\left(3a,a;2a;{\mathrm{e}}^{\ifrac{\mathrm{i}\pi}{3}}\right)=\sqrt{\pi}{% \mathrm{e}}^{\ifrac{\mathrm{i}\pi a}{2}}\frac{2^{2a}\Gamma\left(\frac{1}{2}+a% \right)}{3^{(3a+1)/2}}\left(\frac{1}{\Gamma\left(\frac{1}{3}+a\right)\Gamma% \left(\frac{2}{3}\right)}+\frac{1}{\Gamma\left(\frac{2}{3}+a\right)\Gamma\left% (\frac{1}{3}\right)}\right),$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $a$ : real or complex parameter Proof sketch: From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\ifrac{\mathrm{i}\pi a}{2}}2^{-2a}3^{(3a+1)/2}F\left(3a,a;% 2a;{\mathrm{e}}^{\ifrac{\mathrm{i}\pi}{3}}\right)\Gamma\left(\frac{1}{3}+a% \right)/\Gamma\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$ . Two linearly independent solutions of this difference equation are $w_{1}(a)=1$ and $w_{2}(a)=\Gamma\left(\frac{1}{3}+a\right)/\Gamma\left(\frac{2}{3}+a\right)$ . Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim\sqrt{\pi}\left(\frac{1}{\Gamma\left(\frac{2}{3}\right)}+\frac{a^{-1/3% }}{\Gamma\left(\frac{1}{3}\right)}\right)$ , as $a\to\infty$ . Hence, $w(a)=\sqrt{\pi}\left(\frac{1}{\Gamma\left(\frac{2}{3}\right)}+\frac{w_{2}(a)}{% \Gamma\left(\frac{1}{3}\right)}\right)$ . Notes: See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer. Referenced by: §15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions Permalink: http://dlmf.nist.gov/15.4.E34 Encodings: TeX, pMML, png Addition (effective with 1.1.2): This equation was added. See also: Annotations for §15.4(iii), §15.4 and Ch.15

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) $a=-\frac{1}{3},-\frac{4}{3},-\frac{7}{3},\dots$ , and in (15.4.34) $a=0,-1,-2,\dots$ . Compare the final paragraph in §15.2(ii).

15.3 Graphics 15.5 Derivatives and Contiguous Functions

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