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DLMF: §12.8 Recurrence Relations and Derivatives ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions

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12 Parabolic Cylinder Functions Properties 12.7 Relations to Other Functions 12.9 Asymptotic Expansions for Large Variable

§12.8 Recurrence Relations and Derivatives

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Permalink:: http://dlmf.nist.gov/12.8
See also:: Annotations for Ch.12

Contents

§12.8(i) Recurrence Relations

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Keywords:: parabolic cylinder functions, recurrence relations
Notes:: See Miller (1955, pp. 65).
Permalink:: http://dlmf.nist.gov/12.8.i
See also:: Annotations for §12.8 and Ch.12

12.8.1		$\displaystyle zU\left(a,z\right)-U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a+% 1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.2		$\displaystyle U'\left(a,z\right)+\tfrac{1}{2}zU\left(a,z\right)+(a+\tfrac{1}{2% })U\left(a+1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.3		$\displaystyle U'\left(a,z\right)-\tfrac{1}{2}zU\left(a,z\right)+U\left(a-1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.4		$\displaystyle 2U'\left(a,z\right)+U\left(a-1,z\right)+(a+\tfrac{1}{2})U\left(a% +1,z\right)$	$\displaystyle=0.$
ⓘ Symbols: $U\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12

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

12.8.5		$\displaystyle zV\left(a,z\right)-V\left(a+1,z\right)+(a-\tfrac{1}{2})V\left(a-% 1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.6		$\displaystyle V'\left(a,z\right)-\tfrac{1}{2}zV\left(a,z\right)-(a-\tfrac{1}{2% })V\left(a-1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.7		$\displaystyle V'\left(a,z\right)+\tfrac{1}{2}zV\left(a,z\right)-V\left(a+1,z\right)$	$\displaystyle=0,$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12
12.8.8		$\displaystyle 2V'\left(a,z\right)-V\left(a+1,z\right)-(a-\tfrac{1}{2})V\left(a% -1,z\right)$	$\displaystyle=0.$
ⓘ Symbols: $V\left(\NVar{a},\NVar{z}\right)$ : 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 and Ch.12

§12.8(ii) Derivatives

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Keywords:: derivatives, parabolic cylinder functions
Notes:: (12.8.9), (12.8.10), (12.8.11), and (12.8.12) can be obtained from (12.5.1), (12.5.6), (12.5.7), and (12.5.9), respectively.
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.8.ii
See also:: Annotations for §12.8 and Ch.12

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

12.8.9		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}U\left(a,z% \right)\right)=(-1)^{m}{\left(\tfrac{1}{2}+a\right)_{m}}e^{\frac{1}{4}z^{2}}U% \left(a+m,z\right),$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E9 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.10		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}U\left(a,% z\right)\right)=(-1)^{m}e^{-\frac{1}{4}z^{2}}U\left(a-m,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E10 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.11		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{\frac{1}{4}z^{2}}V\left(a,z% \right)\right)=e^{\frac{1}{4}z^{2}}V\left(a+m,z\right),$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E11 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.8.12		$\frac{{\mathrm{d}}^{m}}{{\mathrm{d}z}^{m}}\left(e^{-\frac{1}{4}z^{2}}V\left(a,% z\right)\right)=(-1)^{m}{\left(\tfrac{1}{2}-a\right)_{m}}e^{-\frac{1}{4}z^{2}}% V\left(a-m,z\right).$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.8.E12 Encodings: TeX, pMML, png See also: Annotations for §12.8(ii), §12.8 and Ch.12

12.7 Relations to Other Functions 12.9 Asymptotic Expansions for Large Variable

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