§8.17 Incomplete Beta Functions

§8.17(i) Definitions and Basic Properties

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

8.17.1
8.17.2

where, as in §5.12, denotes the Beta function:

8.17.4
8.17.5 positive integers; .

Addendum: For a companion equation see (8.17.24).

8.17.6.

For a historical profile of see Dutka (1981).

§8.17(iii) Integral Representation

With , , and ,

where and the branches of and are continuous on the path and assume their principal values when .

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

With

8.17.11
8.17.12
8.17.13
8.17.15

§8.17(v) Continued Fraction

where

The and convergents are less than , and the and convergents are greater than .

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

The expansion (8.17.22) converges rapidly for . For or , more rapid convergence is obtained by computing 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 positive integers; .

Compare (8.17.5).