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
§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 $(b-a)M\left(a-1,b,z\right)+(2a-b+z)M\left(a,b,z\right)-aM\left(a+1,b,z\right)=0,$

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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 : 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 $b(b-1)M\left(a,b-1,z\right)+b(1-b-z)M\left(a,b,z\right)+z(b-a)M\left(a,b+1,z% \right)=0,$

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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 : 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 $(a-b+1)M\left(a,b,z\right)-aM\left(a+1,b,z\right)+(b-1)M\left(a,b-1,z\right)=0,$

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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 : 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 $bM\left(a,b,z\right)-bM\left(a-1,b,z\right)-zM\left(a,b+1,z\right)=0,$

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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 : 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 $b(a+z)M\left(a,b,z\right)+z(a-b)M\left(a,b+1,z\right)-abM\left(a+1,b,z\right)=0,$

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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 : 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 $(a-1+z)M\left(a,b,z\right)+(b-a)M\left(a-1,b,z\right)+(1-b)M\left(a,b-1,z% \right)=0.$

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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 : 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 $U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left(a+1,b,z\right% )=0,$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $(b-a-1)U\left(a,b-1,z\right)+(1-b-z)U\left(a,b,z\right)+zU\left(a,b+1,z\right)% =0,$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $U\left(a,b,z\right)-aU\left(a+1,b,z\right)-U\left(a,b-1,z\right)=0,$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $(b-a)U\left(a,b,z\right)+U\left(a-1,b,z\right)-zU\left(a,b+1,z\right)=0,$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $(a+z)U\left(a,b,z\right)-zU\left(a,b+1,z\right)+a(b-a-1)U\left(a+1,b,z\right)=0,$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $(a-1+z)U\left(a,b,z\right)-U\left(a-1,b,z\right)+(a-b+1)U\left(a,b-1,z\right)=0.$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $(a+1)zM\left(a+2,b+2,z\right)+(b+1)(b-z)M\left(a+1,b+1,z\right)-b(b+1)M\left(a% ,b,z\right)=0,$

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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 : 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 $(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,z\right)-U\left(a,b,z\right)% =0.$

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and : 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 $\frac{\mathrm{d}}{\mathrm{d}z}M\left(a,b,z\right)=\frac{a}{b}M\left(a+1,b+1,z% \right),$

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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 with respect to and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}M\left(a,b,z\right)=\frac{{\left(a% \right)_{n}}}{{\left(b\right)_{n}}}M\left(a+n,b+n,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}M\left(a,b,z% \right)\right)={\left(a\right)_{n}}z^{a+n-1}M\left(a+n,b,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}M\left(a,b,z\right)% \right)={\left(b-n\right)_{n}}z^{b-n-1}M\left(a,b-n,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}M\left(a% ,b,z\right)\right)={\left(b-a\right)_{n}}z^{b-a+n-1}e^{-z}M\left(a-n,b,z\right),$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}M\left(a,b,z\right)% \right)=(-1)^{n}\frac{{\left(b-a\right)_{n}}}{{\left(b\right)_{n}}}e^{-z}M% \left(a,b+n,z\right),$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}M\left(a,b,z% \right)\right)={\left(b-n\right)_{n}}z^{b-n-1}e^{-z}M\left(a-n,b-n,z\right).$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\frac{\mathrm{d}}{\mathrm{d}z}U\left(a,b,z\right)=-aU\left(a+1,b+1,z\right),$

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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 with respect to and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}U\left(a,b,z\right)=(-1)^{n}{\left(a% \right)_{n}}U\left(a+n,b+n,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{a-1}U\left(a,b,z% \right)\right)={\left(a\right)_{n}}{\left(a-b+1\right)_{n}}z^{a+n-1}U\left(a+n% ,b,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}U\left(a,b,z\right)% \right)=(-1)^{n}{\left(a-b+1\right)_{n}}z^{b-n-1}U\left(a,b-n,z\right),$

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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 with respect to , : nonnegative integer and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(z^{b-a-1}e^{-z}U\left(a% ,b,z\right)\right)=(-1)^{n}z^{b-a+n-1}e^{-z}U\left(a-n,b,z\right),$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-z}U\left(a,b,z\right)% \right)=(-1)^{n}e^{-z}U\left(a,b+n,z\right),$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^{b-1}e^{-z}U\left(a,b,z% \right)\right)=(-1)^{n}z^{b-n-1}e^{-z}U\left(a-n,b-n,z\right).$

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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 with respect to , $\mathrm{e}$ : base of natural logarithm, : nonnegative integer and : 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 $\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}=z^{n}\frac{{\mathrm{d}}^{n}}% {{\mathrm{d}z}^{n}}z^{n},$ $n=1,2,3,\dots$ .

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : nonnegative integer and : 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