13 Confluent Hypergeometric FunctionsKummer Functions13.2 Definitions and Basic Properties13.4 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

Contents

1. §13.3(i) Recurrence Relations
2. §13.3(ii) Differentiation Formulas

§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		$(b-a)M(a-1,b,z)+(2a-b+z)M(a,b,z)-aM(a+1,b,z)$	$=0,$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$b(b-1)M(a,b-1,z)+b(1-b-z)M(a,b,z)+z(b-a)M(a,b+1,z)$	$=0,$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$(a-b+1)M(a,b,z)-aM(a+1,b,z)+(b-1)M(a,b-1,z)$	$=0,$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$bM(a,b,z)-bM(a-1,b,z)-zM(a,b+1,z)$	$=0,$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$b(a+z)M(a,b,z)+z(a-b)M(a,b+1,z)-abM(a+1,b,z)$	$=0,$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$(a-1+z)M(a,b,z)+(b-a)M(a-1,b,z)+(1-b)M(a,b-1,z)$	$=0.$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$U(a-1,b,z)+(b-2a-z)U(a,b,z)+a(a-b+1)U(a+1,b,z)$	$=0,$
ⓘ Symbols: $U(a,b,z)$: 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		$(b-a-1)U(a,b-1,z)+(1-b-z)U(a,b,z)+zU(a,b+1,z)$	$=0,$
ⓘ Symbols: $U(a,b,z)$: 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		$U(a,b,z)-aU(a+1,b,z)-U(a,b-1,z)$	$=0,$
ⓘ Symbols: $U(a,b,z)$: 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		$(b-a)U(a,b,z)+U(a-1,b,z)-zU(a,b+1,z)$	$=0,$
ⓘ Symbols: $U(a,b,z)$: 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		$(a+z)U(a,b,z)-zU(a,b+1,z)+a(b-a-1)U(a+1,b,z)$	$=0,$
ⓘ Symbols: $U(a,b,z)$: 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		$(a-1+z)U(a,b,z)-U(a-1,b,z)+(a-b+1)U(a,b-1,z)$	$=0.$
ⓘ Symbols: $U(a,b,z)$: 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		$$(a+1)zM(a+2,b+2,z)+(b+1)(b-z)M(a+1,b+1,z)-b(b+1)M(a,b,z)=0,$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ 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		$$(a+1)zU(a+2,b+2,z)+(z-b)U(a+1,b+1,z)-U(a,b,z)=0.$$
ⓘ Symbols: $U(a,b,z)$: 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		$$\frac{d}{dz}M(a,b,z)=\frac{a}{b}M(a+1,b+1,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}M(a,b,z)=\frac{{\left(a\right)}_n}{{\left(b\right)}_n}M(a+n,b+n,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}M(a,b,z)\right)={\left(a\right)}_n{z}^{a+n-1}M(a+n,b,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left(z^{b-1}M(a,b,z)\right)={\left(b-n\right)}_n{z}^{b-n-1}M(a,b-n,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n\left(z^{b-a-1}{\mathrm{e}}^{-z}M(a,b,z)\right)={\left(b-a\right)}_n{z}^{b-a+n-1}{\mathrm{e}}^{-z}M(a-n,b,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{-z}M(a,b,z)\right)={(-1)}^n\frac{{\left(b-a\right)}_n}{{\left(b\right)}_n}{\mathrm{e}}^{-z}M(a,b+n,z),$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left(z^{b-1}{\mathrm{e}}^{-z}M(a,b,z)\right)={\left(b-n\right)}_n{z}^{b-n-1}{\mathrm{e}}^{-z}M(a-n,b-n,z).$$
ⓘ Symbols: $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\frac{d}{dz}U(a,b,z)=-aU(a+1,b+1,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}U(a,b,z)={(-1)}^n{\left(a\right)}_{n}U(a+n,b+n,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n\left(z^{a-1}U(a,b,z)\right)={\left(a\right)}_n{\left(a-b+1\right)}_n{z}^{a+n-1}U(a+n,b,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left(z^{b-1}U(a,b,z)\right)={(-1)}^n{\left(a-b+1\right)}_n{z}^{b-n-1}U(a,b-n,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n\left(z^{b-a-1}{\mathrm{e}}^{-z}U(a,b,z)\right)={(-1)}^n{z}^{b-a+n-1}{\mathrm{e}}^{-z}U(a-n,b,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left({\mathrm{e}}^{-z}U(a,b,z)\right)={(-1)}^n{\mathrm{e}}^{-z}U(a,b+n,z),$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: 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		$$\frac{d^n}{{dz}^n}\left(z^{b-1}{\mathrm{e}}^{-z}U(a,b,z)\right)={(-1)}^n{z}^{b-n-1}{\mathrm{e}}^{-z}U(a-n,b-n,z).$$
ⓘ Symbols: $U(a,b,z)$: Kummer confluent hypergeometric function, $\frac{df}{dx}$: 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		$${\left(z\frac{d}{dz}z\right)}^n=z^n\frac{d^n}{{dz}^n}{z}^n,$$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\frac{df}{dx}$: 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 Properties13.4 Integral Representations

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