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12 Parabolic Cylinder FunctionsProperties

§12.13 Sums

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§12.13(i) Addition Theorems

12.13.1 U(a,x+y)=e12xy+14y2m=0(-y)mm!U(a-m,x),
12.13.2 U(a,x+y)=e-12xy-14y2m=0(-a-12m)ymU(a+m,x),
12.13.3 V(a,x+y)=e12xy+14y2m=0(a-12m)ymV(a-m,x),
12.13.4 V(a,x+y)=e-12xy-14y2m=0ymm!V(a+m,x).
12.13.5 U(a,xcost+ysint)=e14(xsint-ycost)2m=0(-a-12m)(tant)mU(m+a,x)U(-m-12,y),
a-12,0t14π.
12.13.6 n!U(n+12,z)=ine-12z2erfc(z/2)U(-n-12,iz)+m=112n+12U(2m-n-12,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).