16 Generalized Hypergeometric Functions & Meijer G-Function Generalized Hypergeometric Functions 16.2 Definition and Analytic Properties 16.4 Argument Unity

§16.3 Derivatives and Contiguous Functions

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Permalink:: http://dlmf.nist.gov/16.3
See also:: Annotations for Ch.16

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

§16.3(i) Differentiation Formulas

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Keywords:: derivatives, generalized hypergeometric functions
Notes:: See Luke (1975, §5.2.2). For (16.3.5) see Fleury and Turbiner (1994).
Permalink:: http://dlmf.nist.gov/16.3.i
See also:: Annotations for §16.3 and Ch.16

16.3.1		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p% }\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf{a}\right)_{n}}}{{\left% (\mathbf{b}\right)_{n}}}{{}_{p}F_{q}}\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n% ,\dots,b_{q}+n};z\right),$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized 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 $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.3.E1 Encodings: TeX, pMML, png See also: Annotations for §16.3(i), §16.3 and Ch.16

16.3.2		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma}{{}_{p}F_{q}}\left({% a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={\left(\gamma-n+1% \right)_{n}}z^{\gamma-n}{{}_{p+1}F_{q+1}}\left({\gamma+1,a_{1},\dots,a_{p}% \atop\gamma+1-n,b_{1},\dots,b_{q}};z\right),$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized 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 $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.3.E2 Encodings: TeX, pMML, png See also: Annotations for §16.3(i), §16.3 and Ch.16

16.3.3		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}{{}_{p+1}F_% {q}}\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)={% \left(\gamma\right)_{n}}z^{\gamma+n-1}{{}_{p+1}F_{q}}\left({\gamma+n,a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized 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 $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.3.E3 Encodings: TeX, pMML, png See also: Annotations for §16.3(i), §16.3 and Ch.16

16.3.4		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma-1}{{}_{p}F_{q+1}}% \left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z\right)\right)={\left(% \gamma-n\right)_{n}}z^{\gamma-n-1}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop% \gamma-n,b_{1},\dots,b_{q}};z\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized 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 $x$ , $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.3.E4 Encodings: TeX, pMML, png See also: Annotations for §16.3(i), §16.3 and Ch.16

Other versions of these identities can be constructed with the aid of the operator identity

16.3.5		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$
$n=1,2,\dots$ .
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable Referenced by: §16.3(i) Permalink: http://dlmf.nist.gov/16.3.E5 Encodings: TeX, pMML, png See also: Annotations for §16.3(i), §16.3 and Ch.16

§16.3(ii) Contiguous Functions

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Keywords:: contiguous functions, generalized hypergeometric functions
Notes:: See Rainville (1960, §48).
Permalink:: http://dlmf.nist.gov/16.3.ii
See also:: Annotations for §16.3 and Ch.16

Two generalized hypergeometric functions ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$ , and corresponding parameters differ by integers. If $p\leq q+1$ , then any $q+2$ distinct contiguous functions are linearly related. Examples are provided by the following recurrence relations:

16.3.6		$z{{}_{0}F_{1}}\left(-;b+1;z\right)+b(b-1){{}_{0}F_{1}}\left(-;b;z\right)-b(b-1% ){{}_{0}F_{1}}\left(-;b-1;z\right)=0,$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $z$ : complex variable and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Permalink: http://dlmf.nist.gov/16.3.E6 Encodings: TeX, pMML, png See also: Annotations for §16.3(ii), §16.3 and Ch.16

16.3.7		${{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1}(a_{1}+% 1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1% }\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3}+1)\right)+{{}_{3}F_{2}}\left(% {a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)\left((2a_{1}-b_{1})(2a_{1}-b_{2}% )+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-{{}_{3}F_{2}}\left({a_{1}% -1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{1}-b_{1})(a_{1}-b_{2})=0.$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$ generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/16.3.E7 Encodings: TeX, pMML, png See also: Annotations for §16.3(ii), §16.3 and Ch.16

For further examples see §§13.3(i), 15.5(ii), and the following references: Rainville (1960, §48), Wimp (1968), and Luke (1975, §5.13).

16.2 Definition and Analytic Properties 16.4 Argument Unity

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§16.3 Derivatives and Contiguous Functions

Contents

§16.3(i) Differentiation Formulas

§16.3(ii) Contiguous Functions