12 Parabolic Cylinder Functions Properties 12.12 Integrals 12.14 The Function

W\left(a,x\right)

§12.13 Sums

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Keywords:: parabolic cylinder functions, sums
Permalink:: http://dlmf.nist.gov/12.13
See also:: Annotations for Ch.12

§12.13(i) Addition Theorems

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Keywords:: addition theorems, parabolic cylinder functions
Notes:: (12.13.1)–(12.13.4) follow from the results in §12.8(ii) and Taylor’s theorem (§1.10(i)). For (12.13.5) see Shanker (1939) or Erdélyi et al. (1953b, Chapter 8). For (12.13.6) see Lepe (1985).
Permalink:: http://dlmf.nist.gov/12.13.i
See also:: Annotations for §12.13 and Ch.12

12.13.1		$U\left(a,x y\right)=e^{\frac{1}{2}xy \frac{1}{4}y^{2}}\sum_{m=0}^{\infty}\frac% {(-y)^{m}}{m!}U\left(a-m,x\right),$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter Referenced by: §12.13(i) Permalink: http://dlmf.nist.gov/12.13.E1 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

12.13.2		$U\left(a,x y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{-a-\tfrac{1}{2}}{m}y^{m}U\left(a m,x\right),$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter Permalink: http://dlmf.nist.gov/12.13.E2 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

12.13.3		$V\left(a,x y\right)=e^{\frac{1}{2}xy \frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \genfrac{(}{)}{0.0pt}{}{a-\tfrac{1}{2}}{m}y^{m}V\left(a-m,x\right),$
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\mathrm{e}$ : base of natural logarithm, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter Permalink: http://dlmf.nist.gov/12.13.E3 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

12.13.4		$V\left(a,x y\right)=e^{-\frac{1}{2}xy-\frac{1}{4}y^{2}}\sum_{m=0}^{\infty}% \frac{y^{m}}{m!}V\left(a m,x\right).$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter Referenced by: §12.13(i) Permalink: http://dlmf.nist.gov/12.13.E4 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

12.13.5		$U\left(a,x\cos t y\sin t\right)\\ =e^{\frac{1}{4}(x\sin t-y\cos t)^{2}}\*\sum_{m=0}^{\infty}\genfrac{(}{)}{0.0pt% }{}{-a-\tfrac{1}{2}}{m}(\tan t)^{m}U\left(m a,x\right)U\left(-m-\tfrac{1}{2},y% \right),$
$\Re a\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi$ .
ⓘ Symbols: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $x$ : real variable, $y$ : real variable and $a$ : real or complex parameter Referenced by: §12.13(i) Permalink: http://dlmf.nist.gov/12.13.E5 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

12.13.6		$n!U\left(n \tfrac{1}{2},z\right)=i^{n}e^{-\frac{1}{2}z^{2}}\operatorname{erfc}% (z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right) \sum_{m=1}^{\left\lfloor\frac{1}{% 2}n \frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1}{2},z\right),$
$n=0,1,2,\dots.$
ⓘ Symbols: $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\mathrm{i}$ : imaginary unit, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $z$ : complex variable and $n$ : nonnegative integer Referenced by: §12.13(i) Permalink: http://dlmf.nist.gov/12.13.E6 Encodings: TeX, pMML, png See also: Annotations for §12.13(i), §12.13 and Ch.12

For $\operatorname{erfc}$ see §7.2(i).

§12.13(ii) Other Series

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Permalink:: http://dlmf.nist.gov/12.13.ii
See also:: Annotations for §12.13 and Ch.12

For other series see Dhar (1940), Hansen (1975, pp. 421–422), Hillion (1997), Miller (1974), Prudnikov et al. (1986b, p. 651), Shanker (1940b, a, c), and Varma (1941).

12.12 Integrals 12.14 The Function

W\left(a,x\right)

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§12.13 Sums

Contents

§12.13(i) Addition Theorems

§12.13(ii) Other Series