12 Parabolic Cylinder Functions Properties 12.12 Integrals 12.14 The Function $\mathop{W\/}\nolimits\!\left(a,x\right)$

§12.13 Sums

Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.13

§12.13(i) Addition Theorems
§12.13(ii) Other Series

§12.13(i) Addition Theorems

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).
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.13.i

12.13.1 $\mathop{U\/}\nolimits\!\left(a,x+y\right)=e^{{\frac{1}{2}xy+\frac{1}{4}y^{2}}}% \sum_{{m=0}}^{\infty}\frac{(-y)^{m}}{m!}\mathop{U\/}\nolimits\!\left(a-m,x% \right),$

Symbols:: : base of exponential function, : : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, : real variable and : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E1
Encodings:: TeX, pMML, png

12.13.2 $\mathop{U\/}\nolimits\!\left(a,x+y\right)=e^{{-\frac{1}{2}xy-\frac{1}{4}y^{2}}% }\sum_{{m=0}}^{\infty}\binom{-a-\tfrac{1}{2}}{m}y^{m}\mathop{U\/}\nolimits\!% \left(a+m,x\right),$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, : real variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.13.E2
Encodings:: TeX, pMML, png

12.13.3 $\mathop{V\/}\nolimits\!\left(a,x+y\right)=e^{{\frac{1}{2}xy+\frac{1}{4}y^{2}}}% \sum_{{m=0}}^{\infty}\binom{a-\tfrac{1}{2}}{m}y^{m}\mathop{V\/}\nolimits\!% \left(a-m,x\right),$

Symbols:: $\binom{m}{n}$ : binomial coefficient, : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, : real variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/12.13.E3
Encodings:: TeX, pMML, png

12.13.4 $\mathop{V\/}\nolimits\!\left(a,x+y\right)=e^{{-\frac{1}{2}xy-\frac{1}{4}y^{2}}% }\sum_{{m=0}}^{\infty}\frac{y^{m}}{m!}\mathop{V\/}\nolimits\!\left(a+m,x\right).$

Symbols:: : base of exponential function, : : factorial, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : real variable, : real variable and : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E4
Encodings:: TeX, pMML, png

12.13.5 $\mathop{U\/}\nolimits\!\left(a,x\mathop{\cos\/}\nolimits t+y\mathop{\sin\/}% \nolimits t\right)\\ =e^{{\frac{1}{4}(x\mathop{\sin\/}\nolimits t-y\mathop{\cos\/}\nolimits t)^{2}}% }\*\sum_{{m=0}}^{\infty}\binom{-a-\tfrac{1}{2}}{m}(\mathop{\tan\/}\nolimits t)% ^{m}\mathop{U\/}\nolimits\!\left(m+a,x\right)\mathop{U\/}\nolimits\!\left(-m-% \tfrac{1}{2},y\right),$ $\realpart{a}\leq-\tfrac{1}{2},0\leq t\leq\tfrac{1}{4}\pi$ .

Symbols:: $\binom{m}{n}$ : binomial coefficient, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, : real variable, : real variable and : real or complex parameter
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E5
Encodings:: TeX, pMML, png

12.13.6 $n!\mathop{U\/}\nolimits\!\left(n+\tfrac{1}{2},z\right)=i^{n}e^{{-\frac{1}{2}z^% {2}}}\mathop{\mathrm{erfc}\/}\nolimits(z/\sqrt{2})\mathop{U\/}\nolimits\!\left% (-n-\tfrac{1}{2},iz\right)+\sum_{{m=1}}^{{\left\lfloor\frac{1}{2}n+\frac{1}{2}% \right\rfloor}}\mathop{U\/}\nolimits\!\left(2m-n-\tfrac{1}{2},z\right),$ $n=0,1,2,\dots.$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, : base of exponential function, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : complex variable and : nonnegative integer
Referenced by:: §12.13(i)
Permalink:: http://dlmf.nist.gov/12.13.E6
Encodings:: TeX, pMML, png

For $\mathop{\mathrm{erfc}\/}\nolimits$ see §7.2(i).

§12.13(ii) Other Series

Permalink:: http://dlmf.nist.gov/12.13.ii

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 12.12 Integrals 12.14 The Function $\mathop{W\/}\nolimits\!\left(a,x\right)$

§12.13 Sums

Contents

§12.13(i) Addition Theorems

§12.13(ii) Other Series