I am trying to (numerically) integrate functions of the following form: $$f(p):=p^{\alpha-1}(1-p)^{\beta-1}\prod_{k=0}^{n}\left(1+x_{k+1}p\left(\exp\left\{-2\pi\imath\frac{l}{n+1}\right\}-1\right)\right)$$ where $p\in[0,1]$, $x\in[0,1]^{n+1}$, $l\in\{0,\ldots,n+1\}$, $\alpha,\beta > 0$ and $\imath$ is the imaginary unit in $\mathbb{C}$. This clearly is of the form: $$f(p)=g(p)P[p]$$ where $g:[0,1]\to\mathbb{R}$ is the density of a $\beta$-distributed random variable and $P\in\mathbb{C}[X]$.
Being a pure mathematician myself, I have started to look through numerical methods and am trying to narrow down the methods to use. Aside of the standard methods, there are plenty of sources for numerical integration of special functions. The problem I am facing is the following:
Question 1: is the function $f$ a 'special function', i.e. does the function $f$ have a special name I can look up?
Question 2: Does anybody by chance know which quadrature rule he or she would choose and would be happy to share this knowledge with me?
Question 3: Is there an explicit method (by name) to find a polynomial $Q\in\mathbb{C}[X]$ of appropriate degree which is $g$-orthogonal to the Lagrange polynomials for a given set of quadrature points in $[0,1]$ (related to weighted quadrature methods)?
I wanted to quickly mention the following 'solution' I found after realizing that I can get rid of the product: $$f(p)=\sum_{m=0}^{n+1}c^{m}p^{\alpha+m-1}(1-p)^{\beta-1}\left(\sum_{1\leq i_{1}<\cdots<i_{m}\leq n+1}x_{i_{1}}\cdots x_{i_{m}}\right)$$ where $c:=\exp\left\{-2\pi\imath\frac{l}{n+1}\right\}-1$, so that we obtain: $$\int_{[0,1]}f(p)\operatorname{d}p=\sum_{m=0}^{n+1}c^{m}\left(\sum_{1\leq i_{1}<\cdots<i_{m}\leq n+1}x_{i_{1}}\cdots x_{i_{m}}\right)B(\alpha+m,\beta)$$ where $B$ is the Beta-function. I do not have any problems finding appropriate methods for the latter and all I have to deal with now is to calculate the coefficients at appropriate accuracy.