13 Confluent Hypergeometric Functions Kummer Functions 13.2 Definitions and Basic Properties 13.4 Integral Representations

§13.3 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/13.3
See also:: Annotations for Ch.13

§13.3(i) Recurrence Relations

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Keywords:: Kummer functions, Kummer’s equation, differential equation, equivalent form, recurrence relations
Notes:: See Slater (1960, §2.2). Note that (2.2.7) in this reference contains an error: the correct version is (13.3.13). To see that (13.3.13) and (13.3.14) are equivalent to (13.2.1) use (13.3.16) and (13.3.23).
Referenced by:: §13.29(iv), §16.3(ii)
Permalink:: http://dlmf.nist.gov/13.3.i
See also:: Annotations for §13.3 and Ch.13

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

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Keywords:: Kummer functions, derivatives
Notes:: See Slater (1960, §2.1). Note that (2.1.32) contains an error; the correct version is (13.3.28). For the operator identity (13.3.29) see Fleury and Turbiner (1994).
Referenced by:: §13.10(i), §18.9(iii)
Permalink:: http://dlmf.nist.gov/13.3.ii
See also:: Annotations for §13.3 and Ch.13

13.3.15		$\frac{\mathrm{d}}{\mathrm{d}z}M\left(a,b,z\right)=\frac{a}{b}M\left(a+1,b+1,z% \right),$
ⓘ 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, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable A&S Ref: 13.4.8 Permalink: http://dlmf.nist.gov/13.3.E15 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.16		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}M\left(a,b,z\right)=\frac{{\left(a% \right)_{n}}}{{\left(b\right)_{n}}}M\left(a+n,b+n,z\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $z$ : complex variable A&S Ref: 13.4.9 Referenced by: §13.3(i) Permalink: http://dlmf.nist.gov/13.3.E16 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.17		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}M\left(a,b,z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}M\left(a+n,b,z\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E17 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.18		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}M\left(a,b,z\right)% \right)={\left(b-n\right)_{n}}z^{b-n-1}M\left(a,b-n,z\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $z$ : complex variable Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/13.3.E18 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.19		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}M\left(a% ,b,z\right)\right)={\left(b-a\right)_{n}}z^{b-a+n-1}e^{-z}M\left(a-n,b,z\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E19 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.20		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}M\left(a,b,z\right)% \right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b\right)_{n}}}e^{-z}M% \left(a,b+n,z\right),$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §18.17(iv) Permalink: http://dlmf.nist.gov/13.3.E20 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.21		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}M\left(a,b,z% \right)\right)={\left(b-n\right)_{n}}z^{b-n-1}e^{-z}M\left(a-n,b-n,z\right).$
ⓘ 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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E21 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.22		$\frac{\mathrm{d}}{\mathrm{d}z}U\left(a,b,z\right)=-aU\left(a+1,b+1,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable A&S Ref: 13.4.21 Permalink: http://dlmf.nist.gov/13.3.E22 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.23		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a,b,z\right)=(-1)^{n}{\left(a% \right)_{n}}U\left(a+n,b+n,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent 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 $x$ , $n$ : nonnegative integer and $z$ : complex variable A&S Ref: 13.4.22 Referenced by: §13.3(i) Permalink: http://dlmf.nist.gov/13.3.E23 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.24		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}U\left(a,b,z% \right)\right)={\left(a\right)_{n}}{\left(a-b+1\right)_{n}}z^{a+n-1}U\left(a+n% ,b,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent 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 $x$ , $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E24 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.25		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}U\left(a,b,z\right)% \right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U\left(a,b-n,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent 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 $x$ , $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E25 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.26		$\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}U\left(a% ,b,z\right)\right)=(-1)^{n}z^{b-a+n-1}e^{-z}U\left(a-n,b,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E26 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.27		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}U\left(a,b,z\right)% \right)=(-1)^{n}e^{-z}U\left(a,b+n,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.3.E27 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

13.3.28		$\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}U\left(a,b,z% \right)\right)=(-1)^{n}z^{b-n-1}e^{-z}U\left(a-n,b-n,z\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.3(ii) Permalink: http://dlmf.nist.gov/13.3.E28 Encodings: TeX, pMML, png See also: Annotations for §13.3(ii), §13.3 and Ch.13

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

13.2 Definitions and Basic Properties 13.4 Integral Representations

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§13.3 Recurrence Relations and Derivatives

Contents

§13.3(i) Recurrence Relations

§13.3(ii) Differentiation Formulas