Is this a special function?

Question

Suppose $$ f(z;a) = \int_0^z t^{-a-1}\,(1+t)^{a}\,dt, $$ where $a>1$. Is this function known as a special function? It appears to be close to the following representation of the beta function: $$ B(a',b)=\int_0^\infty\frac{x^{a'-1}}{(1+x)^{a'+b}}\,dx. $$ The only problem is that here it is assumed that $\Re(a')>0$ and $\Re(b)>0$. Otherwise the upper limit of integration can be chopped off at $z$ making it an incomplete beta function with $a'=-a<-1$ and $b=0$, and identical to $f(z;a)$ above. Can one consider it an incomplete beta function?

Answer 1

For $z\ge0$ and $a<0$, we have $~f(z~;a)=(-1)^aB(-z~;-a,~a+1)$. For $a\ge0$, it diverges, due to the integrand's behavior near $0$.