I am asking if this quantity has a name or it is just a real sequence.

Question

Let us consider the Chebychev polynomial function $T_{n}(x)$ where $n$ is a fixed positive integer called the degree and $x$ is the real variable. Let us consider $x$ as a positive integer variable and consider the quantity $T_{x}(x)$. Clearly, this quantity is not a polynomial function. I am asking if it has a name. Or it is just a real sequence.

Answer 1

Unusual but it has an expansion for small values of $x$ $$T_x(x)=1-\frac{\pi ^2 }{8}x^2+\frac{\pi }{2}x^3+\frac{\pi ^4-192}{384} x^4-\frac{\pi \left(\pi ^2-4\right)}{48} x^5+O\left(x^6\right)$$

Edit

If you look at the trigonometric definition of Chebyshev polynomials of the first kind, we have

\begin{cases} T_x(x) =\cos \left(x \cos ^{-1}(x)\right) \qquad & \text{if}\quad |x| \le 1 \\ T_x(x) =\cosh \left(x \cosh ^{-1}(x)\right) \qquad & \text{if}\quad x \ge 1 \\ T_x(x) =(-1)^x \cosh \left(x \cosh ^{-1}(-x)\right) \qquad & \text{if}\quad x \le -1 \end{cases}