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12 Parabolic Cylinder FunctionsProperties

§12.8 Recurrence Relations and Derivatives

Contents
  1. §12.8(i) Recurrence Relations
  2. §12.8(ii) Derivatives

§12.8(i) Recurrence Relations

12.8.1 zU(a,z)U(a1,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(a1,z) =0,
12.8.4 2U(a,z)+U(a1,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)+(a12)V(a1,z) =0,
12.8.6 V(a,z)12zV(a,z)(a12)V(a1,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)(a12)V(a1,z) =0.

§12.8(ii) Derivatives

For m=0,1,2,,

12.8.9 dmdzm(e14z2U(a,z))=(1)m(12+a)me14z2U(a+m,z),
12.8.10 dmdzm(e14z2U(a,z))=(1)me14z2U(am,z),
12.8.11 dmdzm(e14z2V(a,z))=e14z2V(a+m,z),
12.8.12 dmdzm(e14z2V(a,z))=(1)m(12a)me14z2V(am,z).