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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.17 Incomplete Beta Functions

  1. §8.17(i) Definitions and Basic Properties
  2. §8.17(ii) Hypergeometric Representations
  3. §8.17(iii) Integral Representation
  4. §8.17(iv) Recurrence Relations
  5. §8.17(v) Continued Fraction
  6. §8.17(vi) Sums
  7. §8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

§8.17(i) Definitions and Basic Properties

Throughout §§8.17 and 8.18 we assume that a>0, b>0, and 0x1. However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of a, b, and x, and also to complex values.

8.17.1 Bx(a,b)=0xta1(1t)b1dt,
8.17.2 Ix(a,b)=Bx(a,b)/B(a,b),

where, as in §5.12, B(a,b) denotes the beta function:

8.17.3 B(a,b)=Γ(a)Γ(b)Γ(a+b).
8.17.4 Ix(a,b)=1I1x(b,a).
8.17.5 Ix(m,nm+1)=j=mn(nj)xj(1x)nj,
m,n positive integers; 0x<1.

Addendum: For a companion equation see (8.17.24).

8.17.6 Ix(a,a)=12I4x(1x)(a,12),

For a historical profile of Bx(a,b) see Dutka (1981).

§8.17(ii) Hypergeometric Representations

8.17.7 Bx(a,b) =xaaF(a,1b;a+1;x),
8.17.8 Bx(a,b) =xa(1x)baF(a+b,1;a+1;x),
8.17.9 Bx(a,b) =xa(1x)b1aF(1,1ba+1;xx1).

For the hypergeometric function F(a,b;c;z) see §15.2(i).

§8.17(iii) Integral Representation

With a>0, b>0, and 0<x<1,

8.17.10 Ix(a,b)=xa(1x)b2πicic+isa(1s)bdssx,

where x<c<1 and the branches of sa and (1s)b are continuous on the path and assume their principal values when s=c.

Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6.

§8.17(iv) Recurrence Relations


8.17.11 x =1x,
c =a+b1,
8.17.12 Ix(a,b) =xIx(a1,b)+xIx(a,b1),
8.17.13 (a+b)Ix(a,b) =aIx(a+1,b)+bIx(a,b+1),
8.17.14 (a+bx)Ix(a,b)=xbIx(a1,b+1)+aIx(a+1,b),
8.17.15 (b+ax)Ix(a,b)=axIx(a+1,b1)+bIx(a,b+1),
8.17.16 aIx(a+1,b) =(a+cx)Ix(a,b)cxIx(a1,b),
8.17.17 bIx(a,b+1) =(b+cx)Ix(a,b)cxIx(a,b1),
8.17.18 Ix(a,b)=Ix(a+1,b1)+xa(x)b1aB(a,b),
8.17.19 Ix(a,b)=Ix(a1,b+1)xa1(x)bbB(a,b),
8.17.20 Ix(a,b) =Ix(a+1,b)+xa(x)baB(a,b),
8.17.21 Ix(a,b) =Ix(a,b+1)xa(x)bbB(a,b).

§8.17(v) Continued Fraction

8.17.22 Ix(a,b)=xa(1x)baB(a,b)(11+d11+d21+d31+),


8.17.23 d2m =m(bm)x(a+2m1)(a+2m),
d2m+1 =(a+m)(a+b+m)x(a+2m)(a+2m+1).

The 4m and 4m+1 convergents are less than Ix(a,b), and the 4m+2 and 4m+3 convergents are greater than Ix(a,b).

See also Cuyt et al. (2008, pp. 385–389).

The expansion (8.17.22) converges rapidly for x<(a+1)/(a+b+2). For x>(a+1)/(a+b+2) or 1x<(b+1)/(a+b+2), more rapid convergence is obtained by computing I1x(b,a) and using (8.17.4).

§8.17(vi) Sums

For sums of infinite series whose terms involve the incomplete beta function see Hansen (1975, §62).

§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

8.17.24 Ix(m,n)=(1x)nj=m(n+j1j)xj,
m,n positive integers; 0x<1.

Compare (8.17.5).