# §16.3 Derivatives and Contiguous Functions

## §16.3(i) Differentiation Formulas

 16.3.1 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\mathop{{{}_{p}F_{q}}\/}\nolimits\!% \left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)=\frac{{\left(\mathbf% {a}\right)_{n}}}{{\left(\mathbf{b}\right)_{n}}}\mathop{{{}_{p}F_{q}}\/}% \nolimits\!\left({a_{1}+n,\dots,a_{p}+n\atop b_{1}+n,\dots,b_{q}+n};z\right),$
 16.3.2 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma}\mathop{{{}_{p}F_{q}% }\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)\right% )={\left(\gamma-n+1\right)_{n}}z^{\gamma-n}\mathop{{{}_{p+1}F_{q+1}}\/}% \nolimits\!\left({\gamma+1,a_{1},\dots,a_{p}\atop\gamma+1-n,b_{1},\dots,b_{q}}% ;z\right),$
 16.3.3 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{\gamma-1}\mathop{{{}% _{p+1}F_{q}}\/}\nolimits\!\left({\gamma,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{% q}};z\right)\right)={\left(\gamma\right)_{n}}z^{\gamma+n-1}\mathop{{{}_{p+1}F_% {q}}\/}\nolimits\!\left({\gamma+n,a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z% \right),$
 16.3.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{\gamma-1}\mathop{{{}_{p}F_{% q+1}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop\gamma,b_{1},\dots,b_{q}};z% \right)\right)={\left(\gamma-n\right)_{n}}z^{\gamma-n-1}\mathop{{{}_{p}F_{q+1}% }\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop\gamma-n,b_{1},\dots,b_{q}};z% \right).$

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(ii) Contiguous Functions

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

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