# §13.3 Recurrence Relations and Derivatives

## §13.3(i) Recurrence Relations

 13.3.1 $\displaystyle(b-a)M\left(a-1,b,z\right)+(2a-b+z)M\left(a,b,z\right)-aM\left(a+% 1,b,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.1 Permalink: http://dlmf.nist.gov/13.3.E1 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.2 $\displaystyle b(b-1)M\left(a,b-1,z\right)+b(1-b-z)M\left(a,b,z\right)+z(b-a)M% \left(a,b+1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.2 Permalink: http://dlmf.nist.gov/13.3.E2 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.3 $\displaystyle(a-b+1)M\left(a,b,z\right)-aM\left(a+1,b,z\right)+(b-1)M\left(a,b% -1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.3 Permalink: http://dlmf.nist.gov/13.3.E3 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.4 $\displaystyle bM\left(a,b,z\right)-bM\left(a-1,b,z\right)-zM\left(a,b+1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.4 Permalink: http://dlmf.nist.gov/13.3.E4 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.5 $\displaystyle b(a+z)M\left(a,b,z\right)+z(a-b)M\left(a,b+1,z\right)-abM\left(a% +1,b,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.5 Permalink: http://dlmf.nist.gov/13.3.E5 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.6 $\displaystyle(a-1+z)M\left(a,b,z\right)+(b-a)M\left(a-1,b,z\right)+(1-b)M\left% (a,b-1,z\right)$ $\displaystyle=0.$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.6 Permalink: http://dlmf.nist.gov/13.3.E6 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.7 $\displaystyle U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left% (a+1,b,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.15 Permalink: http://dlmf.nist.gov/13.3.E7 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.8 $\displaystyle(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z\right)+zU\left(a% ,b+1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.16 Permalink: http://dlmf.nist.gov/13.3.E8 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.9 $\displaystyle U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left(a,b-1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.17 Permalink: http://dlmf.nist.gov/13.3.E9 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.10 $\displaystyle(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU\left(a,b+1,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.18 Permalink: http://dlmf.nist.gov/13.3.E10 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.11 $\displaystyle(a+z)U\left(a,b,z\right)-zU\left(a,b+1,z\right)+a(b-a-1)U\left(a+% 1,b,z\right)$ $\displaystyle=0,$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.19 Permalink: http://dlmf.nist.gov/13.3.E11 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13 13.3.12 $\displaystyle(a-1+z)U\left(a,b,z\right)-U\left(a-1,b,z\right)+(a-b+1)U\left(a,% b-1,z\right)$ $\displaystyle=0.$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable A&S Ref: 13.4.20 Permalink: http://dlmf.nist.gov/13.3.E12 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13

Kummer’s differential equation (13.2.1) is equivalent to

 13.3.13 $(a+1)zM\left(a+2,b+2,z\right)+(b+1)(b-z)M\left(a+1,b+1,z\right)-b(b+1)M\left(a% ,b,z\right)=0,$ ⓘ Symbols: $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function and $z$: complex variable Referenced by: §13.3(i) Permalink: http://dlmf.nist.gov/13.3.E13 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13

and

 13.3.14 $(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,z\right)-U\left(a,b,z\right)% =0.$ ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function and $z$: complex variable Referenced by: §13.3(i) Permalink: http://dlmf.nist.gov/13.3.E14 Encodings: TeX, pMML, png See also: Annotations for §13.3(i), §13.3 and Ch.13

## §13.3(ii) Differentiation Formulas

 13.3.15 $\frac{\mathrm{d}}{\mathrm{d}z}M\left(a,b,z\right)=\frac{a}{b}M\left(a+1,b+1,z% \right),$
 13.3.16 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}M\left(a,b,z\right)=\frac{{\left(a% \right)_{n}}}{{\left(b\right)_{n}}}M\left(a+n,b+n,z\right),$
 13.3.17 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}M\left(a,b,z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}M\left(a+n,b,z\right),$
 13.3.18 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}M\left(a,b,z\right)% \right)={\left(b-n\right)_{n}}z^{b-n-1}M\left(a,b-n,z\right),$
 13.3.19 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}M\left(a% ,b,z\right)\right)={\left(b-a\right)_{n}}z^{b-a+n-1}e^{-z}M\left(a-n,b,z\right),$
 13.3.20 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}M\left(a,b,z\right)% \right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b\right)_{n}}}e^{-z}M% \left(a,b+n,z\right),$
 13.3.21 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}M\left(a,b,z% \right)\right)={\left(b-n\right)_{n}}z^{b-n-1}e^{-z}M\left(a-n,b-n,z\right).$
 13.3.22 $\frac{\mathrm{d}}{\mathrm{d}z}U\left(a,b,z\right)=-aU\left(a+1,b+1,z\right),$
 13.3.23 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a,b,z\right)=(-1)^{n}{\left(a% \right)_{n}}U\left(a+n,b+n,z\right),$
 13.3.24 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}U\left(a,b,z% \right)\right)={\left(a\right)_{n}}{\left(a-b+1\right)_{n}}z^{a+n-1}U\left(a+n% ,b,z\right),$
 13.3.25 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}U\left(a,b,z\right)% \right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U\left(a,b-n,z\right),$
 13.3.26 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}U\left(a% ,b,z\right)\right)=(-1)^{n}z^{b-a+n-1}e^{-z}U\left(a-n,b,z\right),$
 13.3.27 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}U\left(a,b,z\right)% \right)=(-1)^{n}e^{-z}U\left(a,b+n,z\right),$
 13.3.28 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}U\left(a,b,z% \right)\right)=(-1)^{n}z^{b-n-1}e^{-z}U\left(a-n,b-n,z\right).$

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

 13.3.29 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$ $n=1,2,3,\dots$. ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $n$: nonnegative integer and $z$: complex variable Referenced by: §13.15(ii), §13.3(ii) Permalink: http://dlmf.nist.gov/13.3.E29 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13