§12.8 Recurrence Relations and Derivatives

Permalink:: http://dlmf.nist.gov/12.8

§12.8(i) Recurrence Relations
§12.8(ii) Derivatives

§12.8(i) Recurrence Relations

Notes:: See Miller (1955, pp. 65).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.8.i

12.8.1 $z\mathop{U\/}\nolimits\!\left(a,z\right)-\mathop{U\/}\nolimits\!\left(a-1,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.8.2 ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,z\right)+\tfrac{1}{2}z\mathop{U\/}\nolimits\!\left(a,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.8.3 ${\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,z\right)-\tfrac{1}{2}z\mathop{U\/}\nolimits\!\left(a,z\right)+\mathop{U\/}\nolimits\!\left(a-1,z\right)=0,$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.8.4 $2{\mathop{U\/}\nolimits^{{\prime}}}\!\left(a,z\right)+\mathop{U\/}\nolimits\!\left(a-1,z\right)+(a+\tfrac{1}{2})\mathop{U\/}\nolimits\!\left(a+1,z\right)=0.$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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

12.8.5 $z\mathop{V\/}\nolimits\!\left(a,z\right)-\mathop{V\/}\nolimits\!\left(a+1,z\right)+(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,z\right)=0,$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

12.8.6 ${\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,z\right)-\tfrac{1}{2}z\mathop{V\/}\nolimits\!\left(a,z\right)-(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,z\right)=0,$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

12.8.7 ${\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,z\right)+\tfrac{1}{2}z\mathop{V\/}\nolimits\!\left(a,z\right)-\mathop{V\/}\nolimits\!\left(a+1,z\right)=0,$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

12.8.8 $2{\mathop{V\/}\nolimits^{{\prime}}}\!\left(a,z\right)-\mathop{V\/}\nolimits\!\left(a+1,z\right)-(a-\tfrac{1}{2})\mathop{V\/}\nolimits\!\left(a-1,z\right)=0.$

Symbols:: $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

§12.8(ii) Derivatives

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.
Keywords:: derivatives, parabolic cylinder functions
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.8.ii

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

12.8.9 $\frac{{d}^{m}}{{dz}^{m}}\left(e^{{\frac{1}{4}z^{2}}}\mathop{U\/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}\left(\tfrac{1}{2}+a\right)_{{m}}e^{{\frac{1}{4}z^{2}}}\mathop{U\/}\nolimits\!\left(a+m,z\right),$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.8.10 $\frac{{d}^{m}}{{dz}^{m}}\left(e^{{-\frac{1}{4}z^{2}}}\mathop{U\/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}e^{{-\frac{1}{4}z^{2}}}\mathop{U\/}\nolimits\!\left(a-m,z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

12.8.11 $\frac{{d}^{m}}{{dz}^{m}}\left(e^{{\frac{1}{4}z^{2}}}\mathop{V\/}\nolimits\!\left(a,z\right)\right)=e^{{\frac{1}{4}z^{2}}}\mathop{V\/}\nolimits\!\left(a+m,z\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

12.8.12 $\frac{{d}^{m}}{{dz}^{m}}\left(e^{{-\frac{1}{4}z^{2}}}\mathop{V\/}\nolimits\!\left(a,z\right)\right)=(-1)^{m}\left(\tfrac{1}{2}-a\right)_{{m}}e^{{-\frac{1}{4}z^{2}}}\mathop{V\/}\nolimits\!\left(a-m,z\right).$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $e$ : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

§12.8 Recurrence Relations and Derivatives

Contents

§12.8(i) Recurrence Relations

§12.8(ii) Derivatives