About the Project
12 Parabolic Cylinder FunctionsProperties

§12.13 Sums

Contents
  1. §12.13(i) Addition Theorems
  2. §12.13(ii) Other Series

§12.13(i) Addition Theorems

12.13.1 U(a,x+y)=e12xy+14y2m=0(y)mm!U(am,x),
12.13.2 U(a,x+y)=e12xy14y2m=0(a12m)ymU(a+m,x),
12.13.3 V(a,x+y)=e12xy+14y2m=0(a12m)ymV(am,x),
12.13.4 V(a,x+y)=e12xy14y2m=0ymm!V(a+m,x).
12.13.5 U(a,xcost+ysint)=e14(xsintycost)2m=0(a12m)(tant)mU(m+a,x)U(m12,y),
a12,0t14π.
12.13.6 n!U(n+12,z)=ine12z2erfc(z/2)U(n12,iz)+m=112n+12U(2mn12,z),
n=0,1,2,.

For erfc see §7.2(i).

§12.13(ii) Other Series

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).