§15.5 Derivatives and Contiguous Functions

Permalink:: http://dlmf.nist.gov/15.5

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

§15.5(i) Differentiation Formulas

Notes:: Use Andrews et al. (1999, §2.5) and induction. For (15.5.10) see Fleury and Turbiner (1994).
A&S Ref:: 15.2
Keywords:: derivatives, hypergeometric function
Permalink:: http://dlmf.nist.gov/15.5.i

15.5.1 $\frac{d}{dz}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=\frac{ab}{c}\mathop{F\/}\nolimits\!\left(a+1,b+1;c+1;z\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.2 $\frac{{d}^{n}}{{dz}^{n}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=\frac{\left(a\right)_{{n}}\left(b\right)_{{n}}}{\left(c\right)_{{n}}}\*\mathop{F\/}\nolimits\!\left(a+n,b+n;c+n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.3 $\left(z\frac{d}{dz}z\right)^{n}\left(z^{{a-1}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\left(a\right)_{{n}}z^{{a+n-1}}\mathop{F\/}\nolimits\!\left(a+n,b;c;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.4 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{c-1}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\left(c-n\right)_{{n}}z^{{c-n-1}}\mathop{F\/}\nolimits\!\left(a,b;c-n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.5 $\left(z\frac{d}{dz}z\right)^{n}\left(z^{{c-a-1}}(1-z)^{{a+b-c}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\left(c-a\right)_{{n}}z^{{c-a+n-1}}(1-z)^{{a-n+b-c}}\*\mathop{F\/}\nolimits\!\left(a-n,b;c;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.6 $\frac{{d}^{n}}{{dz}^{n}}\left((1-z)^{{a+b-c}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\frac{\left(c-a\right)_{{n}}\left(c-b\right)_{{n}}}{\left(c\right)_{{n}}}(1-z)^{{a+b-c-n}}\*\mathop{F\/}\nolimits\!\left(a,b;c+n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.7 $\left((1-z)\frac{d}{dz}(1-z)\right)^{n}\left((1-z)^{{a-1}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=(-1)^{n}\frac{\left(a\right)_{{n}}\left(c-b\right)_{{n}}}{\left(c\right)_{{n}}}(1-z)^{{a+n-1}}\*\mathop{F\/}\nolimits\!\left(a+n,b;c+n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.8 $\left((1-z)\frac{d}{dz}(1-z)\right)^{n}\left(z^{{c-1}}(1-z)^{{b-c}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\left(c-n\right)_{{n}}z^{{c-n-1}}(1-z)^{{b-c+n}}\*\mathop{F\/}\nolimits\!\left(a-n,b;c-n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

15.5.9 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{c-1}}(1-z)^{{a+b-c}}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)\right)=\left(c-n\right)_{{n}}z^{{c-n-1}}(1-z)^{{a+b-c-n}}\*\mathop{F\/}\nolimits\!\left(a-n,b-n;c-n;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

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

15.5.10 $\left(z\frac{d}{dz}z\right)^{n}=z^{n}\frac{{d}^{n}}{{dz}^{n}}z^{n},$ $n=1,2,3,\dots$ .

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 Erdélyi et al. (1953a, pp. 102–103).

§15.5(ii) Contiguous Functions

Notes:: See Andrews et al. (1999, §2.5).
A&S Ref:: 15.2
Keywords:: hypergeometric differential equation, hypergeometric function
Referenced by:: §15.19(iv), §16.3(ii), §16.4(iii)
Permalink:: http://dlmf.nist.gov/15.5.ii

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

15.5.11 $(c-a)\mathop{F\/}\nolimits\!\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+a(z-1)\mathop{F\/}\nolimits\!\left(a+1,b;c;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.12 $(b-a)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+a\mathop{F\/}\nolimits\!\left(a+1,b;c;z\right)-b\mathop{F\/}\nolimits\!\left(a,b+1;c;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.13 $(c-a-b)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+a(1-z)\mathop{F\/}\nolimits\!\left(a+1,b;c;z\right)-(c-b)\mathop{F\/}\nolimits\!\left(a,b-1;c;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.14 $c\left(a+(b-c)z\right)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)-ac(1-z)\mathop{F\/}\nolimits\!\left(a+1,b;c;z\right)+(c-a)(c-b)z\mathop{F\/}\nolimits\!\left(a,b;c+1;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.15 $(c-a-1)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+a\mathop{F\/}\nolimits\!\left(a+1,b;c;z\right)-(c-1)\mathop{F\/}\nolimits\!\left(a,b;c-1;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.16 $c(1-z)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)-c\mathop{F\/}\nolimits\!\left(a-1,b;c;z\right)+(c-b)z\mathop{F\/}\nolimits\!\left(a,b;c+1;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.17 $\left(a-1+(b+1-c)z\right)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+(c-a)\mathop{F\/}\nolimits\!\left(a-1,b;c;z\right)-(c-1)(1-z)\mathop{F\/}\nolimits\!\left(a,b;c-1;z\right)=0,$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.18 $c(c-1)(z-1)\mathop{F\/}\nolimits\!\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)+(c-a)(c-b)z\mathop{F\/}\nolimits\!\left(a,b;c+1;z\right)=0.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

By repeated applications of (15.5.11)–(15.5.18) any function $\mathop{F\/}\nolimits\!\left(a+k,b+\ell;c+m;z\right)$ , in which $k,\ell,m$ are integers, can be expressed as a linear combination of $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ 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 ${z(1-z)(a+1)(b+1)}\mathop{F\/}\nolimits\!\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}\mathop{F\/}\nolimits\!\left(a+1,b+1;c+1;z\right)-{c(c+1)}\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=0.$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

Further contiguous relations include:

15.5.20 $z(1-z)\left(\ifrac{d\mathop{F\/}\nolimits\!\left(a,b;c;z\right)}{dz}\right)=(c-a)\mathop{F\/}\nolimits\!\left(a-1,b;c;z\right)+(a-c+bz)\mathop{F\/}\nolimits\!\left(a,b;c;z\right)=(c-b)\mathop{F\/}\nolimits\!\left(a,b-1;c;z\right)+(b-c+az)\mathop{F\/}\nolimits\!\left(a,b;c;z\right),$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

15.5.21 $c(1-z)\left(\ifrac{d\mathop{F\/}\nolimits\!\left(a,b;c;z\right)}{dz}\right)=(c-a)(c-b)\mathop{F\/}\nolimits\!\left(a,b;c+1;z\right)+c(a+b-c)\mathop{F\/}\nolimits\!\left(a,b;c;z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ : 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

§15.5 Derivatives and Contiguous Functions

Contents

§15.5(i) Differentiation Formulas

§15.5(ii) Contiguous Functions