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13 Confluent Hypergeometric FunctionsKummer Functions

§13.3 Recurrence Relations and Derivatives

Contents

§13.3(i) Recurrence Relations

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

§13.3(ii) Differentiation Formulas

13.3.15 zM(a,b,z)=abM(a+1,b+1,z),
13.3.16 nznM(a,b,z)=(a)n(b)nM(a+n,b+n,z),
13.3.17 (zzz)n(za-1M(a,b,z))=(a)nza+n-1M(a+n,b,z),
13.3.18 nzn(zb-1M(a,b,z))=(b-n)nzb-n-1M(a,b-n,z),
13.3.19 (zzz)n(zb-a-1-zM(a,b,z))=(b-a)nzb-a+n-1-zM(a-n,b,z),
13.3.20 nzn(-zM(a,b,z))=(-1)n(b-a)n(b)n-zM(a,b+n,z),
13.3.21 nzn(zb-1-zM(a,b,z))=(b-n)nzb-n-1-zM(a-n,b-n,z).
13.3.22 zU(a,b,z)=-aU(a+1,b+1,z),
13.3.23 nznU(a,b,z)=(-1)n(a)nU(a+n,b+n,z),
13.3.24 (zzz)n(za-1U(a,b,z))=(a)n(a-b+1)nza+n-1U(a+n,b,z),
13.3.25 nzn(zb-1U(a,b,z))=(-1)n(a-b+1)nzb-n-1U(a,b-n,z),
13.3.26 (zzz)n(zb-a-1-zU(a,b,z))=(-1)nzb-a+n-1-zU(a-n,b,z),
13.3.27 nzn(-zU(a,b,z))=(-1)n-zU(a,b+n,z),
13.3.28 nzn(zb-1-zU(a,b,z))=(-1)nzb-n-1-zU(a-n,b-n,z).

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

13.3.29 (zzz)n=znnznzn,
n=1,2,3,.