§13.3 Recurrence Relations and Derivatives

§13.3(i) Recurrence Relations

 13.3.1 $\displaystyle(b-a)\mathop{M\/}\nolimits\!\left(a-1,b,z\right)+(2a-b+z)\mathop{% M\/}\nolimits\!\left(a,b,z\right)-a\mathop{M\/}\nolimits\!\left(a+1,b,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.2 $\displaystyle b(b-1)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)+b(1-b-z)% \mathop{M\/}\nolimits\!\left(a,b,z\right)+z(b-a)\mathop{M\/}\nolimits\!\left(a% ,b+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.3 $\displaystyle(a-b+1)\mathop{M\/}\nolimits\!\left(a,b,z\right)-a\mathop{M\/}% \nolimits\!\left(a+1,b,z\right)+(b-1)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.4 $\displaystyle b\mathop{M\/}\nolimits\!\left(a,b,z\right)-b\mathop{M\/}% \nolimits\!\left(a-1,b,z\right)-z\mathop{M\/}\nolimits\!\left(a,b+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.5 $\displaystyle b(a+z)\mathop{M\/}\nolimits\!\left(a,b,z\right)+z(a-b)\mathop{M% \/}\nolimits\!\left(a,b+1,z\right)-ab\mathop{M\/}\nolimits\!\left(a+1,b,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.6 $\displaystyle(a-1+z)\mathop{M\/}\nolimits\!\left(a,b,z\right)+(b-a)\mathop{M\/% }\nolimits\!\left(a-1,b,z\right)+(1-b)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)$ $\displaystyle=0.$ Symbols: $\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: $=\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$ 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) 13.3.7 $\displaystyle\mathop{U\/}\nolimits\!\left(a-1,b,z\right)+(b-2a-z)\mathop{U\/}% \nolimits\!\left(a,b,z\right)+a(a-b+1)\mathop{U\/}\nolimits\!\left(a+1,b,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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) 13.3.8 $\displaystyle(b-a-1)\mathop{U\/}\nolimits\!\left(a,b-1,z\right)+(1-b-z)\mathop% {U\/}\nolimits\!\left(a,b,z\right)+z\mathop{U\/}\nolimits\!\left(a,b+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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) 13.3.9 $\displaystyle\mathop{U\/}\nolimits\!\left(a,b,z\right)-a\mathop{U\/}\nolimits% \!\left(a+1,b,z\right)-\mathop{U\/}\nolimits\!\left(a,b-1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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) 13.3.10 $\displaystyle(b-a)\mathop{U\/}\nolimits\!\left(a,b,z\right)+\mathop{U\/}% \nolimits\!\left(a-1,b,z\right)-z\mathop{U\/}\nolimits\!\left(a,b+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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) 13.3.11 $\displaystyle(a+z)\mathop{U\/}\nolimits\!\left(a,b,z\right)-z\mathop{U\/}% \nolimits\!\left(a,b+1,z\right)+a(b-a-1)\mathop{U\/}\nolimits\!\left(a+1,b,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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) 13.3.12 $\displaystyle(a-1+z)\mathop{U\/}\nolimits\!\left(a,b,z\right)-\mathop{U\/}% \nolimits\!\left(a-1,b,z\right)+(a-b+1)\mathop{U\/}\nolimits\!\left(a,b-1,z\right)$ $\displaystyle=0.$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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)

Kummer’s differential equation (13.2.1) is equivalent to

 13.3.13 $(a+1)z\mathop{M\/}\nolimits\!\left(a+2,b+2,z\right)+(b+1)(b-z)\mathop{M\/}% \nolimits\!\left(a+1,b+1,z\right)-b(b+1)\mathop{M\/}\nolimits\!\left(a,b,z% \right)=0,$

and

 13.3.14 $(a+1)z\mathop{U\/}\nolimits\!\left(a+2,b+2,z\right)+(z-b)\mathop{U\/}\nolimits% \!\left(a+1,b+1,z\right)-\mathop{U\/}\nolimits\!\left(a,b,z\right)=0.$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: 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(ii) Differentiation Formulas

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

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)