Are there any works on $\int^{z_1}_{z_2}t^{a-1}(1-t)^{b-1}\;f(t) \;dt$?

Question

Beta function is defined as $B(x,y)= \int^{1}_{0} t^{x-1}(1-t)^{y-1} \;dt$. I wonder are there any works on $\int^{z_1}_{z_2}t^{a-1}(1-t)^{b-1}\;f(t) \;dt$ where $f(t)$ is any arbitrary function. If not, is this easily solved? What method would you use to solve this? By part?

Answer 1

If $f$ is the sum of a power series $f(x) = \sum_{n=0}^\infty c_n x^n$ with radius of convergence $>1$, then your integral is $$\sum_{n=0}^\infty c_n B(a+n, b) = \Gamma(b) \sum_{n=0}^\infty c_n \frac{\Gamma(a+n)}{\Gamma(a+b+n)}$$