Can we define the generating function for all $x$ and all $t$

Question

The Chebyshev polynomials of second kind are defined for any $x \in \Bbb R$ (or even $x \in \Bbb C$), e.g. via the recurrence relation $$ U_0(x) = 1 \\ U_1(x) = 2x \\ U_{n+1}(x) = 2x U_n(x) - U_{n-1}(x) $$

The ordinary generating function is

$$∑_{n=0}^{∞}U_{n}(x)t^n=1/(1-2xt+t^2)$$

defined for all $|x|<1$ and $|t|<1$. (https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html)

My question: Since $U_n(x)$ is defined for all reals $x$, then why we need $|t|<1$. Is it related only on the condition $|x|<1$. Can we define the generating function for all $x$ and all $t$.

Answer 1

The generating function can be defined formally. But in that case $x$ and $t$ doesn't represent numbers, they are only formal variables that doesn't take any values, it makes no sense saying "for all $x$ and all $t$".

If one want to consider the generating functions as actual functions with variables that are real (or complex) numbers, then the convergence of the series must be taken into account. $|x|<1$ and $|t|<1$ are sufficient conditions for the series to converge.