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⇑ 13 Confluent Hypergeometric Functions
⇑ Kummer Functions
⇐ 13.2 Definitions and Basic Properties13.4 Integral Representations ⇒

§13.3 Recurrence Relations and Derivatives

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§13.3(i) Recurrence Relations

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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,
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,
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,
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,
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,
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.
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,
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,
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,
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,
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,
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.

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.

§13.3(ii) Differentiation Formulas

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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),
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),
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),
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),
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),
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),
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).
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),
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),
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),
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),
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),
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),
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).

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.
© 2012–2010 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.4; Release date 2012-03-23 Math-Image version (See MathML) . A Printed Companion is available. ⇐ 13.2 Definitions and Basic Properties13.4 Integral Representations ⇒