§12.13 Sums

12.13.1		$$U(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-y)}^m}{m!}U(a-m,x),$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: 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		$$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: 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		$$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: 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		$$V(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{y^m}{m!}V(a+m,x).$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: 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		$$\begin{array}{l}U(a,x\cos t+y\sin t)\\ ={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),\end{array}$$
$\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.
ⓘ Symbols: $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: 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		$$n!U(n+\frac{1}{2},z)={\mathrm{i}}^n{\mathrm{e}}^{-\frac{1}{2}{z}^2}\mathrm{erfc}(z/\sqrt{2})U(-n-\frac{1}{2},\mathrm{i}z)+\sum _{m=1}^{\lfloor \frac{1}{2}n+\frac{1}{2}\rfloor }U(2m-n-\frac{1}{2},z),$$
$n=0,1,2,\mathrm{\dots }.$
ⓘ Symbols: $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\lfloor x\rfloor$: floor of $x$, $\mathrm{i}$: imaginary unit, $U(a,z)$: 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 $\mathrm{erfc}$ see §7.2(i).

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).