Digital Library of Mathematical Functions
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12 Parabolic Cylinder FunctionsProperties

§12.8 Recurrence Relations and Derivatives

Contents

§12.8(i) Recurrence Relations

12.8.1 zU(a,z)-U(a-1,z)+(a+12)U(a+1,z) =0,
12.8.2 U(a,z)+12zU(a,z)+(a+12)U(a+1,z) =0,
12.8.3 U(a,z)-12zU(a,z)+U(a-1,z) =0,
12.8.4 2U(a,z)+U(a-1,z)+(a+12)U(a+1,z) =0.

(12.8.1)–(12.8.4) are also satisfied by U¯(a,z).

12.8.5 zV(a,z)-V(a+1,z)+(a-12)V(a-1,z) =0,
12.8.6 V(a,z)-12zV(a,z)-(a-12)V(a-1,z) =0,
12.8.7 V(a,z)+12zV(a,z)-V(a+1,z) =0,
12.8.8 2V(a,z)-V(a+1,z)-(a-12)V(a-1,z) =0.

§12.8(ii) Derivatives

For m=0,1,2,,

12.8.9 mzm(14z2U(a,z))=(-1)m(12+a)m14z2U(a+m,z),
12.8.10 mzm(-14z2U(a,z))=(-1)m-14z2U(a-m,z),
12.8.11 mzm(14z2V(a,z))=14z2V(a+m,z),
12.8.12 mzm(-14z2V(a,z))=(-1)m(12-a)m-14z2V(a-m,z).