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8 Incomplete Gamma and Related FunctionsIncomplete Gamma Functions

§8.8 Recurrence Relations and Derivatives

8.8.1 γ(a+1,z)=aγ(a,z)-zae-z,
8.8.2 Γ(a+1,z)=aΓ(a,z)+zae-z.

If w(a,z)=γ(a,z) or Γ(a,z), then

8.8.3 w(a+2,z)-(a+1+z)w(a+1,z)+azw(a,z)=0.
8.8.4 zγ*(a+1,z)=γ*(a,z)-e-zΓ(a+1).
8.8.5 P(a+1,z)=P(a,z)-zae-zΓ(a+1),
8.8.6 Q(a+1,z)=Q(a,z)+zae-zΓ(a+1).

For n=0,1,2,,

8.8.7 γ(a+n,z)=(a)nγ(a,z)-zae-zk=0n-1Γ(a+n)Γ(a+k+1)zk,
8.8.8 γ(a,z)=Γ(a)Γ(a-n)γ(a-n,z)-za-1e-zk=0n-1Γ(a)Γ(a-k)z-k,
8.8.9 Γ(a+n,z)=(a)nΓ(a,z)+zae-zk=0n-1Γ(a+n)Γ(a+k+1)zk,
8.8.10 Γ(a,z)=Γ(a)Γ(a-n)Γ(a-n,z)+za-1e-zk=0n-1Γ(a)Γ(a-k)z-k,
8.8.11 P(a+n,z)=P(a,z)-zae-zk=0n-1zkΓ(a+k+1),
8.8.12 Q(a+n,z)=Q(a,z)+zae-zk=0n-1zkΓ(a+k+1).
8.8.13 ddzγ(a,z)=-ddzΓ(a,z)=za-1e-z,
8.8.14 aγ*(a,z)|a=0=-E1(z)-lnz.

For E1(z) see §8.19(i).

For n=0,1,2,,

8.8.15 dndzn(z-aγ(a,z))=(-1)nz-a-nγ(a+n,z),
8.8.16 dndzn(z-aΓ(a,z))=(-1)nz-a-nΓ(a+n,z),
8.8.17 dndzn(ezγ(a,z))=(-1)n(1-a)nezγ(a-n,z),
8.8.18 dndzn(zaezγ*(a,z))=za-nezγ*(a-n,z),
8.8.19 dndzn(ezΓ(a,z))=(-1)n(1-a)nezΓ(a-n,z).