Summation of Legendre polynomials with two power functions

Question

I know there is a generating function of Legendre polynomials, that is $$g(x,t)=1/\sqrt{t^2-2tx+1}=\sum _{n=0}^{\infty }t^n P_n(x),$$when $ \left| t\right| <1 $.

So is there any expression in cloded form for $$\sum _{n=0}^{\infty }{t_1}^n {t_2}^n P_n(x),$$when $ \left| t_1\right| <1 $, and $ \left| t_2\right| <1 $ ? How to get this summation?

Thank you very much!

I do appreciate it if someone can give advice or hint.

Answer 1

Just substitute $t=t_1t_2$ into your original generating function. While the condition $|t_1|<1$, $|t_2|<1$ suffices, you can also use the weaker condition of $|t_1t_2|<1$.