# §13.15 Recurrence Relations and Derivatives

## §13.15(i) Recurrence Relations

 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 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 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 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 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 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 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 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 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 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 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 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 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 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 13

## §13.15(ii) Differentiation Formulas

 13.15.15 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{\frac{1}{2}z}z^{\mu-\frac{1}{% 2}(n+1)}M_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$ 13.15.16 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=\frac{{\left(\frac{1}{2}+\mu-\kappa\right)_{n}}}{{\left(1+2\mu% \right)_{n}}}e^{\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}M_{\kappa-\frac{1}{2}n,% \mu+\frac{1}{2}n}\left(z\right),$ 13.15.17 $\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1% }{2}z}z^{-\kappa-1}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-% \kappa-1}M_{\kappa-n,\mu}\left(z\right),$ 13.15.18 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}{\left(-2\mu\right)_{n}}e^{-\frac{1}{2}z}z^{\mu-\frac{1}% {2}(n+1)}M_{\kappa+\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$ 13.15.19 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{-\mu-\frac{1}{2}}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}\frac{{\left(\frac{1}{2}+\mu+\kappa\right)_{n}}}{{\left(% 1+2\mu\right)_{n}}}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}\*M_{\kappa+\frac% {1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$ 13.15.20 $\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{% 1}{2}z}z^{\kappa-1}M_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle={\left(\tfrac{1}{2}+\mu+\kappa\right)_{n}}e^{-\frac{1}{2}z}z^{% \kappa+n-1}\*M_{\kappa+n,\mu}\left(z\right).$ 13.15.21 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}e^{\frac{1}{2}% z}z^{-\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu+\frac{1}{2}n}\left(z% \right),$ 13.15.22 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{\frac{1}{2}z}z% ^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}{\left(\tfrac{1}{2}-\mu-\kappa\right)_{n}}e^{\frac{1}{2}% z}z^{\mu-\frac{1}{2}(n+1)}\*W_{\kappa-\frac{1}{2}n,\mu-\frac{1}{2}n}\left(z% \right),$ 13.15.23 $\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{\frac{1% }{2}z}z^{-\kappa-1}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle={\left(\tfrac{1}{2}+\mu-\kappa\right)_{n}}{\left(\tfrac{1}{2}-% \mu-\kappa\right)_{n}}e^{\frac{1}{2}z}z^{n-\kappa-1}W_{\kappa-n,\mu}\left(z% \right),$ 13.15.24 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{-\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{-\mu-\frac{1}{2}(n+1)}W_{\kappa+% \frac{1}{2}n,\mu+\frac{1}{2}n}\left(z\right),$ 13.15.25 $\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{-\frac{1}{2}z}% z^{\mu-\frac{1}{2}}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\mu-\frac{1}{2}(n+1)}W_{\kappa+\frac% {1}{2}n,\mu-\frac{1}{2}n}\left(z\right),$ 13.15.26 $\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}\left(e^{-\frac{% 1}{2}z}z^{\kappa-1}W_{\kappa,\mu}\left(z\right)\right)$ $\displaystyle=(-1)^{n}e^{-\frac{1}{2}z}z^{\kappa+n-1}W_{\kappa+n,\mu}\left(z% \right).$

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