§15.5 Derivatives and Contiguous Functions

15.5.1		$$\frac{d}{dz}F(a,b;c;z)=\frac{ab}{c}F(a+1,b+1;c+1;z),$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}F(a,b;c;z)=\frac{{\left(a\right)}_n{\left(b\right)}_n}{{\left(c\right)}_n}F(a+n,b+n;c+n;z).$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}F(a,b;c;z)\right)={\left(a\right)}_n{z}^{a+n-1}F(a+n,b;c;z).$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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.E3 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.4		$$\frac{d^n}{{dz}^n}\left(z^{c-1}F(a,b;c;z)\right)={\left(c-n\right)}_n{z}^{c-n-1}F(a,b;c-n;z).$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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.E4 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.5		$$\begin{array}{l}{\left(z\frac{d}{dz}z\right)}^n\left(z^{c-a-1}{(1-z)}^{a+b-c}F(a,b;c;z)\right)\\ ={\left(c-a\right)}_n{z}^{c-a+n-1}{(1-z)}^{a-n+b-c}F(a-n,b;c;z).\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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.E5 Encodings: TeX, pMML, png See also: Annotations for §15.5(i), §15.5 and Ch.15

15.5.6		$$\frac{d^n}{{dz}^n}\left({(1-z)}^{a+b-c}F(a,b;c;z)\right)=\frac{{\left(c-a\right)}_n{\left(c-b\right)}_n}{{\left(c\right)}_n}{(1-z)}^{a+b-c-n}F(a,b;c+n;z).$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\begin{array}{l}{\left((1-z)\frac{d}{dz}(1-z)\right)}^n\left({(1-z)}^{a-1}F(a,b;c;z)\right)\\ ={(-1)}^n\frac{{\left(a\right)}_n{\left(c-b\right)}_n}{{\left(c\right)}_n}{(1-z)}^{a+n-1}F(a+n,b;c+n;z).\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\begin{array}{l}{\left((1-z)\frac{d}{dz}(1-z)\right)}^n\left(z^{c-1}{(1-z)}^{b-c}F(a,b;c;z)\right)\\ ={\left(c-n\right)}_n{z}^{c-n-1}{(1-z)}^{b-c+n}F(a-n,b;c-n;z).\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\begin{array}{l}\frac{d^n}{{dz}^n}\left(z^{c-1}{(1-z)}^{a+b-c}F(a,b;c;z)\right)\\ ={\left(c-n\right)}_n{z}^{c-n-1}{(1-z)}^{a+b-c-n}F(a-n,b-n;c-n;z).\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\frac{df}{dx}$: 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).

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

15.5.11		$(c-a)F(a-1,b;c;z)+\left(2a-c+(b-a)z\right)F(a,b;c;z)+a(z-1)F(a+1,b;c;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$(b-a)F(a,b;c;z)+aF(a+1,b;c;z)-bF(a,b+1;c;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$(c-a-b)F(a,b;c;z)+a(1-z)F(a+1,b;c;z)-(c-b)F(a,b-1;c;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$c\left(a+(b-c)z\right)F(a,b;c;z)-ac(1-z)F(a+1,b;c;z)+(c-a)(c-b)zF(a,b;c+1;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$(c-a-1)F(a,b;c;z)+aF(a+1,b;c;z)-(c-1)F(a,b;c-1;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$c(1-z)F(a,b;c;z)-cF(a-1,b;c;z)+(c-b)zF(a,b;c+1;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$F(a,b;c;z)-F(a-1,b;c;z)-(b/c)zF(a,b+1;c+1;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$\left(a-1+(b+1-c)z\right)F(a,b;c;z)+(c-a)F(a-1,b;c;z)-(c-1)(1-z)F(a,b;c-1;z)$	$=0,$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$c(c-1)(z-1)F(a,b;c-1;z)+c\left(c-1-(2c-a-b-1)z\right)F(a,b;c;z)+(c-a)(c-b)zF(a,b;c+1;z)$	$=0.$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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 $F(a+k,b+\mathrm{\ell };c+m;z)$, in which $k,\mathrm{\ell },m$ are integers, can be expressed as a linear combination of $F(a,b;c;z)$ 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		$$\begin{array}{l}\begin{array}{l}\begin{array}{l}z(1-z)(a+1)(b+1)F(a+2,b+2;c+2;z)\\ +(c-(a+b+1)z)(c+1)F(a+1,b+1;c+1;z)\end{array}\\ -c(c+1)F(a,b;c;z)\end{array}\\ =0.\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ 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		$$\begin{array}{l}z(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)F(a-1,b;c;z)+(a-c+bz)F(a,b;c;z)\\ =(c-b)F(a,b-1;c;z)+(b-c+az)F(a,b;c;z),\end{array}$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: 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		$$c(1-z)\left(dF(a,b;c;z)/dz\right)=(c-a)(c-b)F(a,b;c+1;z)+c(a+b-c)F(a,b;c;z).$$
ⓘ Symbols: $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_{2}F_1(a,b;c;z)$ Gauss’ hypergeometric function, $\frac{df}{dx}$: 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