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).

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is 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