The incomplete Beta function is defined by $$ B(z,a,b) = \int_0^z dt\; t^{a-1} (1 - t)^{b-1} \ . $$ Suppose that I set $b=0$, and assume $a>0$ as well as $0 < z < 1$. My question is what are the large $a \to \infty$ asymptotics for $$ B(z,a,0) \ ? $$ Mathematica is unable to handle the above series. Also, websites like DLMF only have asymptotic expressions for large $a$ in the case that $b>0$ (and in my case I want $b=0$)
EDIT: See for example this link for an asymptotic series which does not work for the above case
If $0<z<1$, we can write $$ B(z,a,0)=\int_0^z \frac{t^{a-1}}{1-t}dt =\int_0^z t^{a-1}\sum_{n=0}^{\infty}t^n \,dt =\sum_{n=0}^{\infty}\frac{z^{a+n}}{n+a}=z^{a}\Phi(z,1,a), $$ where $$ \Phi(z,s,a)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+a)^s} $$ is the Lerch transcendent. You can find the asymptotic expansion of $\Phi(z,s,a)$ for large $a$ and fixed $z$ and $s$ (including $s=1$) in https://en.wikipedia.org/wiki/Lerch_zeta_function.