§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:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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, $e$ : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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, $e$ : base of exponential function, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

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:: $e$ : base of exponential function, $!$ : $n!$ : factorial, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : 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

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, $e$ : 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, $x$ : 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

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, $e$ : base of exponential function, $!$ : $n!$ : factorial, $\left\lfloor x\right\rfloor$ : floor of $x$ , $\mathop{U\/}\nolimits\!\left(a,z\right)$ : 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

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).

§12.13 Sums

Contents

§12.13(i) Addition Theorems

§12.13(ii) Other Series