13 Confluent Hypergeometric Functions Whittaker Functions 13.14 Definitions and Basic Properties 13.16 Integral Representations

§13.15 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/13.15

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

§13.15(i) Recurrence Relations

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 $(\kappa-\mu-\tfrac{1}{2})\mathop{M_{{\kappa-1,\mu}}\/}\nolimits\!\left(z\right% )+(z-2\kappa)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu+% \tfrac{1}{2})\mathop{M_{{\kappa+1,\mu}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.29
Permalink:: http://dlmf.nist.gov/13.15.E1
Encodings:: TeX, pMML, png

13.15.2 $2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}% \nolimits\!\left(z\right)-(z+2\mu)(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits% \!\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa+\frac{1}{% 2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E2
Encodings:: TeX, pMML, png

13.15.3 $(\kappa-\mu-\tfrac{1}{2})\mathop{M_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}% \nolimits\!\left(z\right)+(1+2\mu)\sqrt{z}\mathop{M_{{\kappa,\mu}}\/}\nolimits% \!\left(z\right)-(\kappa+\mu+\tfrac{1}{2})\mathop{M_{{\kappa+\frac{1}{2},\mu+% \frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E3
Encodings:: TeX, pMML, png

13.15.4 $2\mu\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z% \right)-2\mu\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!% \left(z\right)-\sqrt{z}\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.28
Permalink:: http://dlmf.nist.gov/13.15.E4
Encodings:: TeX, pMML, png

13.15.5 $2\mu(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-2\mu(1+2\mu)% \sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z% \right)-(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa-\frac{1}{2},\mu+% \frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E5
Encodings:: TeX, pMML, png

13.15.6 $2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}% \nolimits\!\left(z\right)+(z-2\mu)(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits% \!\left(z\right)+(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa-\frac{1}{% 2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E6
Encodings:: TeX, pMML, png

13.15.7 $2\mu(1+2\mu)\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}% \nolimits\!\left(z\right)-2\mu(1+2\mu)\mathop{M_{{\kappa,\mu}}\/}\nolimits\!% \left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}\mathop{M_{{\kappa+\frac{1}{2}% ,\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0.$

Symbols:: $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E7
Encodings:: TeX, pMML, png

13.15.8 $\mathop{W_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)-% \sqrt{z}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa-\mu-% \tfrac{1}{2})\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!% \left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E8
Encodings:: TeX, pMML, png

13.15.9 $\mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)-% \sqrt{z}\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa+\mu-% \tfrac{1}{2})\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!% \left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : 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 $2\mu\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)-\sqrt{z}\mathop{W_{{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}\nolimits\!\left(z\right)+\sqrt{z}% \mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E10
Encodings:: TeX, pMML, png

13.15.11 $\mathop{W_{{\kappa+1,\mu}}\/}\nolimits\!\left(z\right)+(2\kappa-z)\mathop{W_{{% \kappa,\mu}}\/}\nolimits\!\left(z\right)+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-% \tfrac{1}{2})\mathop{W_{{\kappa-1,\mu}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
A&S Ref:: 13.4.31
Permalink:: http://dlmf.nist.gov/13.15.E11
Encodings:: TeX, pMML, png

13.15.12 $(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2% }}}\/}\nolimits\!\left(z\right)+2\mu\mathop{W_{{\kappa,\mu}}\/}\nolimits\!% \left(z\right)-(\kappa+\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2}% ,\mu-\frac{1}{2}}}\/}\nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E12
Encodings:: TeX, pMML, png

13.15.13 $(\kappa+\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu-\frac{1}{2% }}}\/}\nolimits\!\left(z\right)-(z+2\mu)\mathop{W_{{\kappa,\mu}}\/}\nolimits\!% \left(z\right)+\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu+\frac{1}{2}}}\/}% \nolimits\!\left(z\right)=0,$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E13
Encodings:: TeX, pMML, png

13.15.14 $(\kappa-\mu-\tfrac{1}{2})\sqrt{z}\mathop{W_{{\kappa-\frac{1}{2},\mu+\frac{1}{2% }}}\/}\nolimits\!\left(z\right)-(z-2\mu)\mathop{W_{{\kappa,\mu}}\/}\nolimits\!% \left(z\right)+\sqrt{z}\mathop{W_{{\kappa+\frac{1}{2},\mu-\frac{1}{2}}}\/}% \nolimits\!\left(z\right)=0.$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker confluent hypergeometric function and : complex variable
Permalink:: http://dlmf.nist.gov/13.15.E14
Encodings:: TeX, pMML, png

§13.15(ii) Differentiation Formulas

Notes:: See Slater (1960, §§2.4, 2.4.1).
Keywords:: Whittaker functions, derivatives
Permalink:: http://dlmf.nist.gov/13.15.ii

13.15.15 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{M% _{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(-2\mu\right% )_{{n}}e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{M_{{\kappa-\frac{1}% {2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E15
Encodings:: TeX, pMML, png

13.15.16 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{% M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\frac{\left(\frac{1}{2}+% \mu-\kappa\right)_{{n}}}{\left(1+2\mu\right)_{{n}}}e^{{\frac{1}{2}z}}z^{{-\mu-% \frac{1}{2}(n+1)}}\mathop{M_{{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}% \nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E16
Encodings:: TeX, pMML, png

13.15.17 $\left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{\frac{1}{2}z}}z^{{-\kappa-1}}% \mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}% +\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{n-\kappa-1}}\mathop{M_{{\kappa-n% ,\mu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E17
Encodings:: TeX, pMML, png

13.15.18 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{% M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(-2\mu% \right)_{{n}}e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{M_{{\kappa+% \frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E18
Encodings:: TeX, pMML, png

13.15.19 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop% {M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\frac{\left(% \frac{1}{2}+\mu+\kappa\right)_{{n}}}{\left(1+2\mu\right)_{{n}}}e^{{-\frac{1}{2% }z}}z^{{-\mu-\frac{1}{2}(n+1)}}\*\mathop{M_{{\kappa+\frac{1}{2}n,\mu+\frac{1}{% 2}n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E19
Encodings:: TeX, pMML, png

13.15.20 $\left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{-\frac{1}{2}z}}z^{{\kappa-1}}% \mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}% +\mu+\kappa\right)_{{n}}e^{{-\frac{1}{2}z}}z^{{\kappa+n-1}}\*\mathop{M_{{% \kappa+n,\mu}}\/}\nolimits\!\left(z\right).$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{M_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E20
Encodings:: TeX, pMML, png

13.15.21 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop{% W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(\tfrac{1}{% 2}+\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}(n+1)}}\*% \mathop{W_{{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E21
Encodings:: TeX, pMML, png

13.15.22 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{W% _{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}\left(\tfrac{1}{2% }-\mu-\kappa\right)_{{n}}e^{{\frac{1}{2}z}}z^{{\mu-\frac{1}{2}(n+1)}}\*\mathop% {W_{{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E22
Encodings:: TeX, pMML, png

13.15.23 $\left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{\frac{1}{2}z}}z^{{-\kappa-1}}% \mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=\left(\tfrac{1}{2}% +\mu-\kappa\right)_{{n}}\left(\tfrac{1}{2}-\mu-\kappa\right)_{{n}}e^{{\frac{1}% {2}z}}z^{{n-\kappa-1}}\mathop{W_{{\kappa-n,\mu}}\/}\nolimits\!\left(z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E23
Encodings:: TeX, pMML, png

13.15.24 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{-\mu-\frac{1}{2}}}\mathop% {W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-\frac{1}{2% }z}}z^{{-\mu-\frac{1}{2}(n+1)}}\mathop{W_{{\kappa+\frac{1}{2}n,\mu+\frac{1}{2}% n}}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E24
Encodings:: TeX, pMML, png

13.15.25 $\frac{{d}^{n}}{{dz}^{n}}\left(e^{{-\frac{1}{2}z}}z^{{\mu-\frac{1}{2}}}\mathop{% W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-\frac{1}{2}% z}}z^{{\mu-\frac{1}{2}(n+1)}}\mathop{W_{{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}% }\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E25
Encodings:: TeX, pMML, png

13.15.26 $\left(z\frac{d}{dz}z\right)^{{n}}\left(e^{{-\frac{1}{2}z}}z^{{\kappa-1}}% \mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)\right)=(-1)^{{n}}e^{{-% \frac{1}{2}z}}z^{{\kappa+n-1}}\mathop{W_{{\kappa+n,\mu}}\/}\nolimits\!\left(z% \right).$

Symbols:: $\mathop{W_{{\kappa,\mu}}\/}\nolimits\!\left(z\right)$ : Whittaker 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/13.15.E26
Encodings:: TeX, pMML, png

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

© 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. 13.14 Definitions and Basic Properties 13.16 Integral Representations

§13.15 Recurrence Relations and Derivatives

Contents

§13.15(i) Recurrence Relations

§13.15(ii) Differentiation Formulas