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

§13.15 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/13.15
See also:: Annotations for Ch.13

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

§13.15(i) Recurrence Relations

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Keywords:: Whittaker functions, 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.
Referenced by:: §13.29(iv)
Permalink:: http://dlmf.nist.gov/13.15.i
See also:: Annotations for §13.15 and Ch.13

13.15.1	$\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z\right)+(z-2% \kappa)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{\kappa+1,\mu}% \left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable A&S Ref: 13.4.29 Permalink: http://dlmf.nist.gov/13.15.E1 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.2	$\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(% z\right)-(z+2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2}% )\sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Referenced by: §13.15(i) Permalink: http://dlmf.nist.gov/13.15.E2 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.3	$\displaystyle(\kappa-\mu-\tfrac{1}{2})M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}% \left(z\right)+(1+2\mu)\sqrt{z}M_{\kappa,\mu}\left(z\right)-(\kappa+\mu+\tfrac% {1}{2})M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E3 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.4	$\displaystyle 2\mu M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-2\mu M% _{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}M_{\kappa,\mu}% \left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable A&S Ref: 13.4.28 Permalink: http://dlmf.nist.gov/13.15.E4 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.5	$\displaystyle 2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)-2\mu(1+2\mu)\sqrt{z}M_{% \kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(\kappa-\mu-\tfrac{1}{2})% \sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E5 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.6	$\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(% z\right)+(z-2\mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2}% )\sqrt{z}M_{\kappa-\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E6 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.7	$\displaystyle 2\mu(1+2\mu)\sqrt{z}M_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(% z\right)-2\mu(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})% \sqrt{z}M_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0.$
ⓘ Symbols: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E7 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.8	$\displaystyle W_{\kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)-\sqrt{z}W_{% \kappa,\mu}\left(z\right)+(\kappa-\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu+% \frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E8 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.9	$\displaystyle W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-\sqrt{z}W_{% \kappa,\mu}\left(z\right)+(\kappa+\mu-\tfrac{1}{2})W_{\kappa-\frac{1}{2},\mu-% \frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : 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 See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.10	$\displaystyle 2\mu W_{\kappa,\mu}\left(z\right)-\sqrt{z}W_{\kappa+\frac{1}{2},% \mu+\frac{1}{2}}\left(z\right)+\sqrt{z}W_{\kappa+\frac{1}{2},\mu-\frac{1}{2}}% \left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Referenced by: §13.15(i) Permalink: http://dlmf.nist.gov/13.15.E10 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.11	$\displaystyle W_{\kappa+1,\mu}\left(z\right)+(2\kappa-z)W_{\kappa,\mu}\left(z% \right)+(\kappa-\mu-\tfrac{1}{2})(\kappa+\mu-\tfrac{1}{2})W_{\kappa-1,\mu}% \left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable A&S Ref: 13.4.31 Permalink: http://dlmf.nist.gov/13.15.E11 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.12	$\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{% 1}{2}}\left(z\right)+2\mu W_{\kappa,\mu}\left(z\right)-(\kappa+\mu-\tfrac{1}{2% })\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E12 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.13	$\displaystyle(\kappa+\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu-\frac{% 1}{2}}\left(z\right)-(z+2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+% \frac{1}{2},\mu+\frac{1}{2}}\left(z\right)$	$\displaystyle=0,$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E13 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13
13.15.14	$\displaystyle(\kappa-\mu-\tfrac{1}{2})\sqrt{z}W_{\kappa-\frac{1}{2},\mu+\frac{% 1}{2}}\left(z\right)-(z-2\mu)W_{\kappa,\mu}\left(z\right)+\sqrt{z}W_{\kappa+% \frac{1}{2},\mu-\frac{1}{2}}\left(z\right)$	$\displaystyle=0.$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E14 Encodings: TeX, pMML, png See also: Annotations for §13.15(i), §13.15 and Ch.13

§13.15(ii) Differentiation Formulas

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Keywords:: Whittaker functions, derivatives
Notes:: See Slater (1960, §§2.4, 2.4.1).
Permalink:: http://dlmf.nist.gov/13.15.ii
See also:: Annotations for §13.15 and Ch.13

13.15.15	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{% 2}(n+1)}M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E15 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.16	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu% \right)_{n}}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac{1}{2}n,% \mu+\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E16 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.17	$\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1% }{2}z}z^{-\kappa-1}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-% \kappa-1}M_{\kappa-n,\mu}\left(z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E17 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.18	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{-\frac{1}{2}z}z^{\mu-\frac{1}% {2}(n+1)}M_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E18 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.19	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(% 1+2\mu\right)_{n}}}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*M_{\kappa+\frac% {1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E19 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.20	$\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{% 1}{2}z}z^{\kappa-1}M_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle={\left(\tfrac{1}{2}+\mu+\kappa\right)_{n}}e^{-\frac{1}{2}z}z^{% \kappa+n-1}\*M_{\kappa+n,\mu}\left(z\right).$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E20 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.21	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}% z}z^{-\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z% \right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E21 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.22	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e^{\frac{1}{2}% z}z^{\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z% \right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E22 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.23	$\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1% }{2}z}z^{-\kappa-1}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}{\left(\tfrac{1}{2}-% \mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}W_{\kappa-n,\mu}\left(z% \right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E23 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.24	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}W_{\kappa+% \frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E24 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.25	$\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}W_{\kappa+\frac% {1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E25 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13
13.15.26	$\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{% 1}{2}z}z^{\kappa-1}W_{\kappa,\mu}\left(z\right)\right)$	$\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\kappa+n-1}W_{\kappa+n,\mu}\left(z% \right).$
ⓘ Symbols: $W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $n$ : nonnegative integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.15.E26 Encodings: TeX, pMML, png See also: Annotations for §13.15(ii), §13.15 and Ch.13

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

13.14 Definitions and Basic Properties 13.16 Integral Representations

NIST

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§13.15 Recurrence Relations and Derivatives

Contents

§13.15(i) Recurrence Relations

§13.15(ii) Differentiation Formulas