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⇑ 13 Confluent Hypergeometric Functions
⇑ Whittaker Functions
⇐ 13.14 Definitions and Basic Properties13.16 Integral Representations ⇒

§13.15 Recurrence Relations and Derivatives

Contents

§13.15(i) Recurrence Relations

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Notes:
See Slater (1960, §2.5). Note that (2.5.4) and (2.5.10) contain errors: the correct versions are (13.15.2) and (13.15.10), respectively.
Keywords:
Whittaker functions
Referenced by:
§13.29(iv)
Permalink:
http://dlmf.nist.gov/13.15.i
13.15.1 (\kappa-\mu-\tfrac{1}{2})\mathop{M_{{\kappa-1,\mu}}\/}\nolimits\!\left(z\right)+(z-2\kappa)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\mathop{M_{{\kappa+1,\mu}}\/}\nolimits\!\left(z\right)=0,
13.15.2 2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-(z+2\mu)(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.3 (\kappa-\mu-\tfrac{1}{2})\mathop{M_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)+(1+2\mu)\sqrt{z}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-(\kappa+\mu+\tfrac{1}{2})\mathop{M_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.4 2\mu\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-2\mu\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-\sqrt{z}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=0,
13.15.5 2\mu(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.6 2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)+(z-2\mu)(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.7 2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-2\mu(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0.
13.15.8 \mathop{W_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)-\sqrt{z}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa-\mu-\tfrac{1}{2})\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.9 \mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-\sqrt{z}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu-\tfrac{1}{2})\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
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Symbols:
\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right): Whittaker confluent hypergeometric function and z: complex variable
A&S Ref:
13.4.30 (in different form)
Permalink:
http://dlmf.nist.gov/13.15.E9
Encodings:
TeX, pMML, png
13.15.10 2\mu\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)+\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.11 \mathop{W_{{\kappa+1,\mu}}\/}\nolimits\!\left(z\right)+(2\kappa-z)\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})\mathop{W_{{\kappa-1,\mu}}\/}\nolimits\!\left(z\right)=0,
13.15.12 (\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)+2\mu\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.13 (\kappa+\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-(z+2\mu)\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,
13.15.14 (\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)-(z-2\mu)\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0.

§13.15(ii) Differentiation Formulas

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Notes:
See Slater (1960, §§2.4, 2.4.1).
Keywords:
Whittaker functions, derivatives
Permalink:
http://dlmf.nist.gov/13.15.ii
13.15.15 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(-2\mu\right)_{{n}}e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{M_{{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.16 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\frac{\left(\frac{1}{2}+\mu-\kappa\right)_{{n}}}{\left(1+2\mu\right)_{{n}}}e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}(n+1)}}\mathop{M_{{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.17 \left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{\frac{1}{2}z}}z^{{-\kappa-1}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}+\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{n-\kappa-1}}\mathop{M_{{\kappa-n,\mu}}\/}\nolimits\!\left(z\right),
13.15.18 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(-2\mu\right)_{{n}}e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{M_{{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.19 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\frac{\left(\frac{1}{2}+\mu+\kappa\right)_{{n}}}{\left(1+2\mu\right)_{{n}}}e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}(n+1)}}\*\mathop{M_{{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.20 \left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{-\frac{1}{2}z}}z^{{\kappa-1}}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}+\mu+\kappa\right)_{{n}}e^{{-\frac{1}{2}z}}z^{{\kappa+n-1}}\*\mathop{M_{{\kappa+n,\mu}}\/}\nolimits\!\left(z\right).
13.15.21 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(\tfrac{1}{2}+\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}(n+1)}}\*\mathop{W_{{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.22 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(\tfrac{1}{2}-\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\*\mathop{W_{{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.23 \left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{\frac{1}{2}z}}z^{{-\kappa-1}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}+\mu-\kappa\right)_{{n}}\left(\tfrac{1}{2}-\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{n-\kappa-1}}\mathop{W_{{\kappa-n,\mu}}\/}\nolimits\!\left(z\right),
13.15.24 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}(n+1)}}\mathop{W_{{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.25 \frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{W_{{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),
13.15.26 \left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{-\frac{1}{2}z}}z^{{\kappa-1}}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-\frac{1}{2}z}}z^{{\kappa+n-1}}\mathop{W_{{\kappa+n,\mu}}\/}\nolimits\!\left(z\right).

Other versions of several of the identities in this subsection can be constructed by use of (13.3.29).

© 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. ⇐ 13.14 Definitions and Basic Properties13.16 Integral Representations ⇒