Digital Library of Mathematical Functions
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12 Parabolic Cylinder FunctionsProperties

§12.7 Relations to Other Functions

Contents

§12.7(i) Hermite Polynomials

For the notation see §18.3.

12.7.1 U(-12,z)=D0(z)=-14z2,
12.7.2 U(-n-12,z)=Dn(z)=-14z2Hen(z)=2-n/2-14z2Hn(z/2),
n=0,1,2, ,
12.7.3 V(n+12,z)=2/π14z2(-)nHen(z)=2/π14z2(-)n2-12nHn(z/2),
n=0,1,2,.

§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

For the notation see §§7.2 and 7.18.

12.7.4 V(-12,z)=(2/π)14z2F(z/2),
12.7.5 U(12,z)=D-1(z)=12π14z2erfc(z/2),
12.7.6 U(n+12,z)=D-n-1(z)=π2(-1)nn!-14z2n(12z2erfc(z/2))zn,
n=0,1,2,,
12.7.7 U(n+12,z)=14z2Hhn(z)=π 212(n-1)14z2inerfc(z/2),
n=-1,0,1,.

§12.7(iii) Modified Bessel Functions

For the notation see §10.25(ii).

12.7.8 U(-2,z)=z5/242π(2K14(14z2)+3K34(14z2)-K54(14z2)),
12.7.9 U(-1,z) =z3/222π(K14(14z2)+K34(14z2)),
12.7.10 U(0,z) =z2πK14(14z2),
12.7.11 U(1,z) =z3/22π(K34(14z2)-K14(14z2)).

For these, the corresponding results for U(a,z) with a=2, ±3, -12, -32, -52, and the corresponding results for V(a,z) with a=0, ±1, ±2, ±3, 12, 32, 52, see Miller (1955, pp. 42–43 and 77–79).

§12.7(iv) Confluent Hypergeometric Functions

For the notation see §§13.2(i) and 13.14(i).

The even and odd solutions of (12.2.2) (see (12.4.3)–(12.4.6)) are given by

12.7.12 u1(a,z)=-14z2M(12a+14,12,12z2)=14z2M(-12a+14,12,-12z2),
12.7.13 u2(a,z)=z-14z2M(12a+34,32,12z2)=z14z2M(-12a+34,32,-12z2).

Also,

12.7.14 U(a,z)=2-14-12a-14z2U(12a+14,12,12z2)=2-34-12az-14z2U(12a+34,32,12z2)=2-12az-12W-12a,±14(12z2).

(It should be observed that the functions on the right-hand sides of (12.7.14) are multivalued; hence, for example, z cannot be replaced simply by -z.)