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13 Confluent Hypergeometric FunctionsKummer Functions

§13.3 Recurrence Relations and Derivatives

Contents
  1. §13.3(i) Recurrence Relations
  2. §13.3(ii) Differentiation Formulas

§13.3(i) Recurrence Relations

13.3.1 (ba)M(a1,b,z)+(2ab+z)M(a,b,z)aM(a+1,b,z) =0,
13.3.2 b(b1)M(a,b1,z)+b(1bz)M(a,b,z)+z(ba)M(a,b+1,z) =0,
13.3.3 (ab+1)M(a,b,z)aM(a+1,b,z)+(b1)M(a,b1,z) =0,
13.3.4 bM(a,b,z)bM(a1,b,z)zM(a,b+1,z) =0,
13.3.5 b(a+z)M(a,b,z)+z(ab)M(a,b+1,z)abM(a+1,b,z) =0,
13.3.6 (a1+z)M(a,b,z)+(ba)M(a1,b,z)+(1b)M(a,b1,z) =0.
13.3.7 U(a1,b,z)+(b2az)U(a,b,z)+a(ab+1)U(a+1,b,z) =0,
13.3.8 (ba1)U(a,b1,z)+(1bz)U(a,b,z)+zU(a,b+1,z) =0,
13.3.9 U(a,b,z)aU(a+1,b,z)U(a,b1,z) =0,
13.3.10 (ba)U(a,b,z)+U(a1,b,z)zU(a,b+1,z) =0,
13.3.11 (a+z)U(a,b,z)zU(a,b+1,z)+a(ba1)U(a+1,b,z) =0,
13.3.12 (a1+z)U(a,b,z)U(a1,b,z)+(ab+1)U(a,b1,z) =0.

Kummer’s differential equation (13.2.1) is equivalent to

13.3.13 (a+1)zM(a+2,b+2,z)+(b+1)(bz)M(a+1,b+1,z)b(b+1)M(a,b,z)=0,

and

13.3.14 (a+1)zU(a+2,b+2,z)+(zb)U(a+1,b+1,z)U(a,b,z)=0.

§13.3(ii) Differentiation Formulas

13.3.15 ddzM(a,b,z)=abM(a+1,b+1,z),
13.3.16 dndznM(a,b,z)=(a)n(b)nM(a+n,b+n,z),
13.3.17 (zddzz)n(za1M(a,b,z))=(a)nza+n1M(a+n,b,z),
13.3.18 dndzn(zb1M(a,b,z))=(bn)nzbn1M(a,bn,z),
13.3.19 (zddzz)n(zba1ezM(a,b,z))=(ba)nzba+n1ezM(an,b,z),
13.3.20 dndzn(ezM(a,b,z))=(1)n(ba)n(b)nezM(a,b+n,z),
13.3.21 dndzn(zb1ezM(a,b,z))=(bn)nzbn1ezM(an,bn,z).
13.3.22 ddzU(a,b,z)=aU(a+1,b+1,z),
13.3.23 dndznU(a,b,z)=(1)n(a)nU(a+n,b+n,z),
13.3.24 (zddzz)n(za1U(a,b,z))=(a)n(ab+1)nza+n1U(a+n,b,z),
13.3.25 dndzn(zb1U(a,b,z))=(1)n(ab+1)nzbn1U(a,bn,z),
13.3.26 (zddzz)n(zba1ezU(a,b,z))=(1)nzba+n1ezU(an,b,z),
13.3.27 dndzn(ezU(a,b,z))=(1)nezU(a,b+n,z),
13.3.28 dndzn(zb1ezU(a,b,z))=(1)nzbn1ezU(an,bn,z).

Other versions of several of the identities in this subsection can be constructed with the aid of the operator identity

13.3.29 (zddzz)n=zndndznzn,
n=1,2,3,.