The generating function $G(t, x)$ for the Legendre polynomials $P_n(t)$ is
\(G(t, x)=\frac{1}{\sqrt{1-2 x t+x^2}}=\sum_{n=0}^{\infty} x^n P_n(t),|x|<1\)
If the function f(x) is defined by the integral equation \(\int_0^x f\left(x^{\prime}\right) d x^{\prime}=x G(1, x)\), it can be expressed as
1
\(\sum_{n, m=0}^{\infty} x^{n+m} P_n(1) P_m\left(\frac{1}{2}\right)\)
2
\(\sum_{n, m=0}^{\infty} x^{n+m} P_n(1) P_m(1)\)
3
\(\sum_{n, m=0}^{\infty} x^{n-m} P_n(1) P_m(1)\)
4
\(\sum_{n, m=0}^{\infty} x^{n-m} P_n(0) P_m(1)\)