Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. 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.

where, as in §5.12, $\mathrm{B}(a,b)$ denotes the beta function:

Addendum: For a companion equation see (8.17.24).

For a historical profile of ${\mathrm{B}}_x(a,b)$ see Dutka (1981).

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

With $a>0$, $b>0$, and ,

where and the branches of $s^{-a}$ and ${(1-s)}^{-b}$ 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.

With

where

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

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

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

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

Compare (8.17.5).