15 Hypergeometric Function Properties 15.4 Special Cases 15.6 Integral Representations

§15.5 Derivatives and Contiguous Functions

ⓘ

Permalink:: http://dlmf.nist.gov/15.5
See also:: Annotations for Ch.15

§15.5(i) Differentiation Formulas
§15.5(ii) Contiguous Functions

§15.5(i) Differentiation Formulas

ⓘ

Keywords:: derivatives, hypergeometric function
A&S Ref:: 15.2
Notes:: Use Andrews et al. (1999, §2.5) and induction. For (15.5.10) see Fleury and Turbiner (1994).
Referenced by:: §18.9(iii)
Permalink:: http://dlmf.nist.gov/15.5.i
See also:: Annotations for §15.5 and Ch.15

15.5.1		$\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1% ;c+1;z\right),$
ⓘ 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, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E1 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.2		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a% \right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z% \right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E2 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.3		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/15.5.E3 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.4		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)% \right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/15.5.E4 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.5		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F% \left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*% F\left(a-n,b;c;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/15.5.E5 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.6		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left((1-z)^{a+b-c}F\left(a,b;c;z% \right)\right)=\frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c% \right)_{n}}}(1-z)^{a+b-c-n}\*F\left(a,b;c+n;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E6 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.7		$\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left((1-z)^{a-1}F% \left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(a\right)_{n}}{\left(c-b\right% )_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F\left(a+n,b;c+n;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E7 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.8		$\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b% -c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{b-c+n}\*% F\left(a-n,b;c-n;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E8 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.9		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}(1-z)^{a+b-c}F\left(a,b% ;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{a+b-c-n}\*F\left(a-n,% b-n;c-n;z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E9 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity

15.5.10		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$
$n=1,2,3,\dots$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $z$ : complex variable Referenced by: §15.5(i) Permalink: http://dlmf.nist.gov/15.5.E10 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

See Erdélyi et al. (1953a, pp. 102–103).

§15.5(ii) Contiguous Functions

ⓘ

Keywords:: contiguous, equivalent equation for contiguous functions, hypergeometric differential equation, hypergeometric function, recurrence relations
A&S Ref:: 15.2
Notes:: See Andrews et al. (1999, §2.5).
Referenced by:: §15.19(iv), (15.4.34), §16.3(ii), §16.4(iii)
Permalink:: http://dlmf.nist.gov/15.5.ii
Addition (effective with 1.1.0):: The contiguous relation (15.5.16_5) was added.
See also:: Annotations for §15.5 and Ch.15

The six functions $F\left(a\pm 1,b;c;z\right)$ , $F\left(a,b\pm 1;c;z\right)$ , $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$ .

15.5.11	$\displaystyle(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c% ;z\right)+a(z-1)F\left(a+1,b;c;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.4(i), §15.5(ii) Permalink: http://dlmf.nist.gov/15.5.E11 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.12	$\displaystyle(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1% ;c;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E12 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.13	$\displaystyle(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F% \left(a,b-1;c;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E13 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.14	$\displaystyle c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;% c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E14 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.15	$\displaystyle(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left% (a,b;c-1;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.4(iii) Permalink: http://dlmf.nist.gov/15.5.E15 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.16	$\displaystyle c(1-z)F\left(a,b;c;z\right)-cF\left(a-1,b;c;z\right)+(c-b)zF% \left(a,b;c+1;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E16 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.16_5	$\displaystyle F\left(a,b;c;z\right)-F\left(a-1,b;c;z\right)-(\ifrac{b}{c})zF% \left(a,b+1;c+1;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.5(ii), Erratum (V1.1.0) for Additions Permalink: http://dlmf.nist.gov/15.5.E16_5 Encodings: TeX, pMML, png Addition (effective with 1.1.0): This equation was added. See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.17	$\displaystyle\left(a-1+(b+1-c)z\right)F\left(a,b;c;z\right)+(c-a)F\left(a-1,b;% c;z\right)-(c-1)(1-z)F\left(a,b;c-1;z\right)$	$\displaystyle=0,$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E17 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15
15.5.18	$\displaystyle c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right% )}F\left(a,b;c;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)$	$\displaystyle=0.$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Referenced by: §15.19(iv), §15.5(ii) Permalink: http://dlmf.nist.gov/15.5.E18 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15

By repeated applications of (15.5.11)–(15.5.18) any function $F\left(a+k,b+\ell;c+m;z\right)$ , in which $k,\ell,m$ are integers, can be expressed as a linear combination of $F\left(a,b;c;z\right)$ and any one of its contiguous functions, with coefficients that are rational functions of $a, b, c$ , and $z$ .

An equivalent equation to the hypergeometric differential equation (15.10.1) is

15.5.19		${z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}F\left(a+1,b+% 1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0.$
ⓘ 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, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E19 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15

Further contiguous relations include:

15.5.20		$z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F% \left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z% \right)+(b-c+az)F\left(a,b;c;z\right),$
ⓘ 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, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E20 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15

15.5.21		$c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)(% c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;z\right).$
ⓘ 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, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter Permalink: http://dlmf.nist.gov/15.5.E21 Encodings: TeX, pMML, png See also: Annotations for §15.5(ii), §15.5 and Ch.15

15.4 Special Cases 15.6 Integral Representations

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§15.5 Derivatives and Contiguous Functions

Contents

§15.5(i) Differentiation Formulas

§15.5(ii) Contiguous Functions