In the literature, there are several orthogonal polynomials, like Hermite Polynomials, Legendre Polynomials. However, I would like to ask whether there are any explicit formulas for the integration of several Orthogonal polynomials. For example, I want to calculate the integration $$\int_{-1}^1\prod_{i=1}^k L_i(x)w(x)dx $$ where, $L_i(x)$ is Legendre polynomials and $w(x) = 1$ is the corresponding weight function, $k$ is the number of polynomials. I want to know how to calculate the integration when the number of polynomials $k=3,4$, especially for $k=4$. I want to know there are some explicit expression for integration. Are there tables/references where I can find the results? Besides, I also want to know the integration in the cases of Hermite, Laguerre and Chebyshev polynomials.
Thank you in advance!
The families of orthogonal polynomials you list are all examples of classical orthogonal polynomials. They all have explicit expressions for their coefficients so you can write them out and do the integration explicitly.
For Legendre we have $$ P_n(x) = \sum_{m=0}^n \binom{n}{m}\binom{\frac{n+m-1}{2},n}x^m, $$ so we get $$ \int_{-1}^{1} dx \prod_{n=0}^k P_n(x) = \prod_{n=0}^k \sum_{m=0}^n \frac{1-(-1)^{m+1}}{m+1} \binom{n}{m} \binom{(n+m-1)/2 }{n}. $$