15 Hypergeometric Function Properties 15.4 Special Cases 15.6 Integral Representations

§15.5 Derivatives and Contiguous Functions

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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

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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 $\frac{\mathrm{d}}{\mathrm{d}z}F\left(a,b;c;z\right)=\frac{ab}{c}F\left(a+1,b+1% ;c+1;z\right),$

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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 with respect to , : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}F\left(a,b;c;z\right)=\frac{{\left(a% \right)_{n}}{\left(b\right)_{n}}}{{\left(c\right)_{n}}}\*F\left(a+n,b+n;c+n;z% \right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}F\left(a,b;c;z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}F\left(a+n,b;c;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}F\left(a,b;c;z\right)% \right)={\left(c-n\right)_{n}}z^{c-n-1}F\left(a,b;c-n;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{c-a-1}(1-z)^{a+b-c}F% \left(a,b;c;z\right)\right)={\left(c-a\right)_{n}}z^{c-a+n-1}(1-z)^{a-n+b-c}\*% F\left(a-n,b;c;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left((1-z)^{a+b-c}F\left(a,b;c;z% \right)\right)=\frac{{\left(c-a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c% \right)_{n}}}(1-z)^{a+b-c-n}\*F\left(a,b;c+n;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left((1-z)^{a-1}F% \left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(a\right)_{n}}{\left(c-b\right% )_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F\left(a+n,b;c+n;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)\right)^{n}\left(z^{c-1}(1-z)^{b% -c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{b-c+n}\*% F\left(a-n,b;c-n;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{c-1}(1-z)^{a+b-c}F\left(a,b% ;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}(1-z)^{a+b-c-n}\*F\left(a-n,% b-n;c-n;z\right).$

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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 with respect to , : integer, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$ $n=1,2,3,\dots$ .

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : integer and : 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

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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 $(c-a)F\left(a-1,b;c;z\right)+\left(2a-c+(b-a)z\right)F\left(a,b;c;z\right)+a(z% -1)F\left(a+1,b;c;z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $(b-a)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-bF\left(a,b+1;c;z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $(c-a-b)F\left(a,b;c;z\right)+a(1-z)F\left(a+1,b;c;z\right)-(c-b)F\left(a,b-1;c% ;z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $c\left(a+(b-c)z\right)F\left(a,b;c;z\right)-ac(1-z)F\left(a+1,b;c;z\right)+(c-% a)(c-b)zF\left(a,b;c+1;z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $(c-a-1)F\left(a,b;c;z\right)+aF\left(a+1,b;c;z\right)-(c-1)F\left(a,b;c-1;z% \right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $c(1-z)F\left(a,b;c;z\right)-cF\left(a-1,b;c;z\right)+(c-b)zF\left(a,b;c+1;z% \right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $F\left(a,b;c;z\right)-F\left(a-1,b;c;z\right)-(\ifrac{b}{c})zF\left(a,b+1;c+1;% z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $\left(a-1+(b+1-c)z\right)F\left(a,b;c;z\right)+(c-a)F\left(a-1,b;c;z\right)-(c% -1)(1-z)F\left(a,b;c-1;z\right)=0,$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $c(c-1)(z-1)F\left(a,b;c-1;z\right)+{c\left(c-1-(2c-a-b-1)z\right)}F\left(a,b;c% ;z\right)+(c-a)(c-b)zF\left(a,b;c+1;z\right)=0.$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 , and .

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

15.5.19 ${z(1-z)(a+1)(b+1)}F\left(a+2,b+2;c+2;z\right)+{(c-(a+b+1)z)(c+1)}F\left(a+1,b+% 1;c+1;z\right)-{c(c+1)}F\left(a,b;c;z\right)=0.$

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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, : complex variable, : real or complex parameter, : real or complex parameter and : 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 $z(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)F% \left(a-1,b;c;z\right)+(a-c+bz)F\left(a,b;c;z\right)=(c-b)F\left(a,b-1;c;z% \right)+(b-c+az)F\left(a,b;c;z\right),$

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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 with respect to , : complex variable, : real or complex parameter, : real or complex parameter and : 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 $c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right)}{\mathrm{d}z}\right)=(c-a)(% c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;z\right).$

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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 with respect to , : complex variable, : real or complex parameter, : real or complex parameter and : 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

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§15.5 Derivatives and Contiguous Functions

Contents

§15.5(i) Differentiation Formulas

§15.5(ii) Contiguous Functions