Generating function of orthogonal polynomial basis

Question

I'm studying the bases made up by orthogonal polynomial such as: Hermite, Legendre, Laguerre, Chebyshev. On my book there is a theoretical introduction that gives the difinition of generating function and its relationship with the "classical orthogonal polynomials". In particular that definition states the generic expression: $$ g(x,t)=\sum_{n=0}^{\infty}\frac{p_n(x)}{n!}t^n \\[5ex] p_n(x)=\left[ \frac{d^n}{dt^n} g(x,t)\right]_{t=0}$$ While trying to solve some example referred to Hermite polynomials, was all good, but when I was asked to solve excercises referred to Legendre polynomial, I noticed that something was wrong; indeed I've fastly discovered that for $g(x,t)=(1-2xt+t^2)^{-\frac{1}{2}}$, the correct formula had to be: $$p_n(x)=\frac{1}{n!}\left[ \frac{d^n}{dt^n} g(x,t)\right]_{t=0}$$ After a bit of search on Wikipedia, I discovered that the formula on my book was holding only for Hermite polynomial. Instead, for the other polynomials I found (they are respectively Legendre's, Laguerre's and Chebyshev's cases): $$ \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}P_n(x)t^n \\[5ex] \frac{1}{1-t}e^{-tx/(1-t)}=\sum_{n=0}^{\infty}L_n(x)t^n \\[5ex] \frac{1-tx}{1-2tx+t^2}=\sum_{n=0}^{\infty}T_n(x)t^n \\[5ex]$$ In all of those formulas, does not appear the $n!$ at denominator, and it seems be compatible with the formula I gave; so my question is: there was a typo (or a lack of generality) in the initial introduction? Or there is something I don't know? Can anyone add something to explaination? And lastly... my book state also that for Laguerre's case, the generating function should be: $g(x,t)=\frac{1}{1-t}e^{tx/(1-t)}$ (without the minus at the exponent).. is that a typo?