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

§13.3 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/13.3

§13.3(i) Recurrence Relations
§13.3(ii) Differentiation Formulas

§13.3(i) 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).
Keywords:: Kummer functions, Kummer’s equation
Referenced by:: §13.29(iv), §16.3(ii)
Permalink:: http://dlmf.nist.gov/13.3.i

13.3.1 $(b-a)\mathop{M\/}\nolimits\!\left(a-1,b,z\right)+(2a-b+z)\mathop{M\/}\nolimits% \!\left(a,b,z\right)-a\mathop{M\/}\nolimits\!\left(a+1,b,z\right)=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.2 $b(b-1)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)+b(1-b-z)\mathop{M\/}% \nolimits\!\left(a,b,z\right)+z(b-a)\mathop{M\/}\nolimits\!\left(a,b+1,z\right% )=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.3 $(a-b+1)\mathop{M\/}\nolimits\!\left(a,b,z\right)-a\mathop{M\/}\nolimits\!\left% (a+1,b,z\right)+(b-1)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.4 $b\mathop{M\/}\nolimits\!\left(a,b,z\right)-b\mathop{M\/}\nolimits\!\left(a-1,b% ,z\right)-z\mathop{M\/}\nolimits\!\left(a,b+1,z\right)=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.5 $b(a+z)\mathop{M\/}\nolimits\!\left(a,b,z\right)+z(a-b)\mathop{M\/}\nolimits\!% \left(a,b+1,z\right)-ab\mathop{M\/}\nolimits\!\left(a+1,b,z\right)=0,$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.6 $(a-1+z)\mathop{M\/}\nolimits\!\left(a,b,z\right)+(b-a)\mathop{M\/}\nolimits\!% \left(a-1,b,z\right)+(1-b)\mathop{M\/}\nolimits\!\left(a,b-1,z\right)=0.$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.7 $\mathop{U\/}\nolimits\!\left(a-1,b,z\right)+(b-2a-z)\mathop{U\/}\nolimits\!% \left(a,b,z\right)+a(a-b+1)\mathop{U\/}\nolimits\!\left(a+1,b,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.8 $(b-a-1)\mathop{U\/}\nolimits\!\left(a,b-1,z\right)+(1-b-z)\mathop{U\/}% \nolimits\!\left(a,b,z\right)+z\mathop{U\/}\nolimits\!\left(a,b+1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.9 $\mathop{U\/}\nolimits\!\left(a,b,z\right)-a\mathop{U\/}\nolimits\!\left(a+1,b,% z\right)-\mathop{U\/}\nolimits\!\left(a,b-1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.10 $(b-a)\mathop{U\/}\nolimits\!\left(a,b,z\right)+\mathop{U\/}\nolimits\!\left(a-% 1,b,z\right)-z\mathop{U\/}\nolimits\!\left(a,b+1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.11 $(a+z)\mathop{U\/}\nolimits\!\left(a,b,z\right)-z\mathop{U\/}\nolimits\!\left(a% ,b+1,z\right)+a(b-a-1)\mathop{U\/}\nolimits\!\left(a+1,b,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

13.3.12 $(a-1+z)\mathop{U\/}\nolimits\!\left(a,b,z\right)-\mathop{U\/}\nolimits\!\left(% a-1,b,z\right)+(a-b+1)\mathop{U\/}\nolimits\!\left(a,b-1,z\right)=0.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : 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

Kummer’s differential equation (13.2.1) is equivalent to

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

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
Referenced by:: §13.3(i)
Permalink:: http://dlmf.nist.gov/13.3.E13
Encodings:: TeX, pMML, png

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:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function and : complex variable
Referenced by:: §13.3(i)
Permalink:: http://dlmf.nist.gov/13.3.E14
Encodings:: TeX, pMML, png

§13.3(ii) Differentiation Formulas

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).
Keywords:: Kummer functions, derivatives
Referenced by:: §13.10(i)
Permalink:: http://dlmf.nist.gov/13.3.ii

13.3.15 $\frac{d}{dz}\mathop{M\/}\nolimits\!\left(a,b,z\right)=\frac{a}{b}\mathop{M\/}% \nolimits\!\left(a+1,b+1,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\frac{df}{dx}$ : 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

13.3.16 $\frac{{d}^{n}}{{dz}^{n}}\mathop{M\/}\nolimits\!\left(a,b,z\right)=\frac{\left(% a\right)_{{n}}}{\left(b\right)_{{n}}}\mathop{M\/}\nolimits\!\left(a+n,b+n,z% \right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

13.3.17 $\left(z\frac{d}{dz}z\right)^{{n}}\left(z^{{a-1}}\mathop{M\/}\nolimits\!\left(a% ,b,z\right)\right)=\left(a\right)_{{n}}z^{{a+n-1}}\mathop{M\/}\nolimits\!\left% (a+n,b,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E17
Encodings:: TeX, pMML, png

13.3.18 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{b-1}}\mathop{M\/}\nolimits\!\left(a,b,z% \right)\right)=\left(b-n\right)_{{n}}z^{{b-n-1}}\mathop{M\/}\nolimits\!\left(a% ,b-n,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E18
Encodings:: TeX, pMML, png

13.3.19 $\left(z\frac{d}{dz}z\right)^{{n}}\left(z^{{b-a-1}}e^{{-z}}\mathop{M\/}% \nolimits\!\left(a,b,z\right)\right)=\left(b-a\right)_{{n}}z^{{b-a+n-1}}e^{{-z% }}\mathop{M\/}\nolimits\!\left(a-n,b,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E19
Encodings:: TeX, pMML, png

13.3.20 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-z}}\mathop{M\/}\nolimits\!\left(a,b,z\right% )\right)=(-1)^{{n}}\frac{\left(b-a\right)_{{n}}}{\left(b\right)_{{n}}}e^{{-z}}% \mathop{M\/}\nolimits\!\left(a,b+n,z\right),$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E20
Encodings:: TeX, pMML, png

13.3.21 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{b-1}}e^{{-z}}\mathop{M\/}\nolimits\!\left(a,% b,z\right)\right)=\left(b-n\right)_{{n}}z^{{b-n-1}}e^{{-z}}\mathop{M\/}% \nolimits\!\left(a-n,b-n,z\right).$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E21
Encodings:: TeX, pMML, png

13.3.22 $\frac{d}{dz}\mathop{U\/}\nolimits\!\left(a,b,z\right)=-a\mathop{U\/}\nolimits% \!\left(a+1,b+1,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\frac{df}{dx}$ : 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

13.3.23 $\frac{{d}^{n}}{{dz}^{n}}\mathop{U\/}\nolimits\!\left(a,b,z\right)=(-1)^{{n}}% \left(a\right)_{{n}}\mathop{U\/}\nolimits\!\left(a+n,b+n,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : 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

13.3.24 $\left(z\frac{d}{dz}z\right)^{{n}}\left(z^{{a-1}}\mathop{U\/}\nolimits\!\left(a% ,b,z\right)\right)=\left(a\right)_{{n}}\left(a-b+1\right)_{{n}}z^{{a+n-1}}% \mathop{U\/}\nolimits\!\left(a+n,b,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E24
Encodings:: TeX, pMML, png

13.3.25 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{b-1}}\mathop{U\/}\nolimits\!\left(a,b,z% \right)\right)=(-1)^{{n}}\left(a-b+1\right)_{{n}}z^{{b-n-1}}\mathop{U\/}% \nolimits\!\left(a,b-n,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of with respect to , : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E25
Encodings:: TeX, pMML, png

13.3.26 $\left(z\frac{d}{dz}z\right)^{{n}}\left(z^{{b-a-1}}e^{{-z}}\mathop{U\/}% \nolimits\!\left(a,b,z\right)\right)=(-1)^{{n}}z^{{b-a+n-1}}e^{{-z}}\mathop{U% \/}\nolimits\!\left(a-n,b,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E26
Encodings:: TeX, pMML, png

13.3.27 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-z}}\mathop{U\/}\nolimits\!\left(a,b,z\right% )\right)=(-1)^{{n}}e^{{-z}}\mathop{U\/}\nolimits\!\left(a,b+n,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Permalink:: http://dlmf.nist.gov/13.3.E27
Encodings:: TeX, pMML, png

13.3.28 $\frac{{d}^{n}}{{dz}^{n}}\left(z^{{b-1}}e^{{-z}}\mathop{U\/}\nolimits\!\left(a,% b,z\right)\right)=(-1)^{{n}}z^{{b-n-1}}e^{{-z}}\mathop{U\/}\nolimits\!\left(a-% n,b-n,z\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\frac{df}{dx}$ : derivative of with respect to , : base of exponential function, : nonnegative integer and : complex variable
Referenced by:: §13.3(ii)
Permalink:: http://dlmf.nist.gov/13.3.E28
Encodings:: TeX, pMML, png

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

13.3.29 $\left(z\frac{d}{dz}z\right)^{n}=z^{n}\frac{{d}^{n}}{{dz}^{n}}z^{n},$ $n=1,2,3,\dots$ .

Symbols:: $\frac{df}{dx}$ : 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

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 13.2 Definitions and Basic Properties 13.4 Integral Representations

§13.3 Recurrence Relations and Derivatives

Contents

§13.3(i) Recurrence Relations

§13.3(ii) Differentiation Formulas