What is the combinatorics meaning of the generating function for Legendre polynomials?

Question

I know the generating function has been a super useful tool when finding the Legendre polynomials (or other special functions), or even used to estimate the static electric potential. In the Physics textbook, it is directly given as: $$\frac{1}{\sqrt{1-2ht+h^2}}=\sum_{i=0}^\infty h^iP_i(t)$$ I am super confused after taking a Combinatorics course cause I can't find a word relevent to this in my textbook, but I am quite sure there must be some combinatorics argument underlies any generating function.

(p.s. first time asking question here, sorry for my poor English)

Answer 1

I like to view at this from a historical location:

see Why do the Legendre Polynomials have these coefficients?

So - for me the generating function is clear and the way that it is a sum of polynomial functions, for me is more the definition of the Legendre polynomials - which then are subject to very nice recurrence, integral and differential equations.