§16.3 Derivatives and Contiguous Functions

Permalink:: http://dlmf.nist.gov/16.3

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

§16.3(i) Differentiation Formulas

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

16.3.1 $\frac{{d}^{n}}{{dz}^{n}}\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},% \dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac{\left(\mathbf{a}\right)_{{n% }}}{\left(\mathbf{b}\right)_{{n}}}\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left% ({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n,\dots,b_{q}+n};z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : nonnegative integer, : 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
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16.3.2 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{\gamma}\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits% \!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right)=\left(% \gamma-n+1\right)_{{n}}z^{{\gamma-n}}\mathop{{{}_{{p+1}}F_{{q+1}}}\/}\nolimits% \!\left({\gamma+1,a_{1},\dots,a_{p}\atop\gamma+1-n,b_{1},\dots,b_{q}};z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : nonnegative integer, : 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

16.3.3 $\left(z\frac{d}{dz}z\right)^{n}\left(z^{{\gamma-1}}\mathop{{{}_{{p+1}}F_{{q}}}% \/}\nolimits\!\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)% \right)=\left(\gamma\right)_{{n}}z^{{\gamma+n-1}}\mathop{{{}_{{p+1}}F_{{q}}}\/% }\nolimits\!\left({\gamma+n,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right),$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : nonnegative integer, : 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

16.3.4 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{\gamma-1}}\mathop{{{}_{{p}}F_{{q+1}}}\/}% \nolimits\!\left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z\right)% \right)=\left(\gamma-n\right)_{{n}}z^{{\gamma-n-1}}\mathop{{{}_{{p}}F_{{q+1}}}% \/}\nolimits\!\left({a_{1},\dots,a_{p}\atop\gamma-n,b_{1},\dots,b_{q}};z\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer, : nonnegative integer, : 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

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

16.3.5 $\left(z\frac{d}{dz}z\right)^{n}=z^{n}\frac{{d}^{n}}{{dz}^{n}}z^{n},$ $n=1,2,\dots$ .

Symbols:: $\frac{df}{dx}$ : derivative of with respect to and : complex variable
Referenced by:: §16.3(i)
Permalink:: http://dlmf.nist.gov/16.3.E5
Encodings:: TeX, pMML, png

§16.3(ii) Contiguous Functions

Notes:: See Rainville (1960, §48).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/16.3.ii

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

16.3.6 $z\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits\!\left(-;b+1;z\right)+b(b-1)\mathop{{% {}_{{0}}F_{{1}}}\/}\nolimits\!\left(-;b;z\right)-b(b-1)\mathop{{{}_{{0}}F_{{1}% }}\/}\nolimits\!\left(-;b-1;z\right)=0,$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : 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

16.3.7 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a_{1}+2,a_{2},a_{3}\atop b_{1},b% _{2}};z\right)a_{1}(a_{1}+1)(1-z)+\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left% ({a_{1}+1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)a_{1}\left(b_{1}+b_{2}-3a_{1}-% 2+z(2a_{1}-a_{2}-a_{3}+1)\right)+\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left(% {a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)\left((2a_{1}-b_{1})(2a_{1}-b_{2}% )+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})\right)-\mathop{{{}_{{3}}F_{{2}}}% \/}\nolimits\!\left({a_{1}-1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{1}-b_{1% })(a_{1}-b_{2})=0.$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, : 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

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

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§16.3 Derivatives and Contiguous Functions

Contents

§16.3(i) Differentiation Formulas

§16.3(ii) Contiguous Functions