# §12.8 Recurrence Relations and Derivatives

## §12.8(i) Recurrence Relations

 12.8.1 $\displaystyle z\mathop{U\/}\nolimits\!\left(a,z\right)-\mathop{U\/}\nolimits\!% \left(a-1,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.4 Referenced by: §12.8(i) Permalink: http://dlmf.nist.gov/12.8.E1 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.2 $\displaystyle\mathop{U\/}\nolimits'\!\left(a,z\right)+\tfrac{1}{2}z\mathop{U\/% }\nolimits\!\left(a,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,% z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.1 Permalink: http://dlmf.nist.gov/12.8.E2 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.3 $\displaystyle\mathop{U\/}\nolimits'\!\left(a,z\right)-\tfrac{1}{2}z\mathop{U\/% }\nolimits\!\left(a,z\right)+\mathop{U\/}\nolimits\!\left(a-1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.2 Permalink: http://dlmf.nist.gov/12.8.E3 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.4 $\displaystyle 2\mathop{U\/}\nolimits'\!\left(a,z\right)+\mathop{U\/}\nolimits% \!\left(a-1,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,z\right)$ $\displaystyle=0.$ Symbols: $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.3 Referenced by: §12.8(i) Permalink: http://dlmf.nist.gov/12.8.E4 Encodings: TeX, pMML, png See also: Annotations for 12.8(i)

(12.8.1)–(12.8.4) are also satisfied by $\mathop{\overline{U}\/}\nolimits\!\left(a,z\right)$.

 12.8.5 $\displaystyle z\mathop{V\/}\nolimits\!\left(a,z\right)-\mathop{V\/}\nolimits\!% \left(a+1,z\right)+(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.8 Permalink: http://dlmf.nist.gov/12.8.E5 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.6 $\displaystyle\mathop{V\/}\nolimits'\!\left(a,z\right)-\tfrac{1}{2}z\mathop{V\/% }\nolimits\!\left(a,z\right)-(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,% z\right)$ $\displaystyle=0,$ Symbols: $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.5 Permalink: http://dlmf.nist.gov/12.8.E6 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.7 $\displaystyle\mathop{V\/}\nolimits'\!\left(a,z\right)+\tfrac{1}{2}z\mathop{V\/% }\nolimits\!\left(a,z\right)-\mathop{V\/}\nolimits\!\left(a+1,z\right)$ $\displaystyle=0,$ Symbols: $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.6 Permalink: http://dlmf.nist.gov/12.8.E7 Encodings: TeX, pMML, png See also: Annotations for 12.8(i) 12.8.8 $\displaystyle 2\mathop{V\/}\nolimits'\!\left(a,z\right)-\mathop{V\/}\nolimits% \!\left(a+1,z\right)-(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,z\right)$ $\displaystyle=0.$ Symbols: $\mathop{V\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.6.7 Permalink: http://dlmf.nist.gov/12.8.E8 Encodings: TeX, pMML, png See also: Annotations for 12.8(i)

## §12.8(ii) Derivatives

For $m=0,1,2,\dots$,

 12.8.9 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}\mathop{U% \/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}+a\right)_{m}% }e^{\frac{1}{4}z^{2}}\mathop{U\/}\nolimits\!\left(a+m,z\right),$
 12.8.10 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}\mathop{U% \/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}e^{-\frac{1}{4}z^{2}}\mathop{U\/% }\nolimits\!\left(a-m,z\right),$
 12.8.11 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}\mathop{V% \/}\nolimits\!\left(a,z\right)\right)=e^{\frac{1}{4}z^{2}}\mathop{V\/}% \nolimits\!\left(a+m,z\right),$
 12.8.12 $\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}\mathop{V% \/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}-a\right)_{m}% }e^{-\frac{1}{4}z^{2}}\mathop{V\/}\nolimits\!\left(a-m,z\right).$