Let $M_n(x) = x^n$ be the standard monomials. The binomial formula allows one to expand $M_n(ax+b)$ as a linear combination of $M_k(x)$, for $k \leq n$, giving

$$ M_n(ax+b) = (ax+b)^n = \sum_{k=0}^n \binom{n}{k} a^k x^k b^{n-k} = \sum_{k=0}^n \binom{n}{k} a^k b^{n-k} M_k(x). $$

**Question:** Do equivalent formulas exist for other polynomial sequences? I am mostly interested in the Chebyshev polynomials of the first kind, $T_n(x) = \cos(n \arccos(x))$, seeking a formula of the form

$$ T_n(ax+b) = \sum_{k=0}^{n} C(n, k, a, b) T_k(x). $$

I believe such a formula ought to exist, because one could always transform $T_n(x)$ into monomial basis, expand terms using the binomial formula, then transform back into a linear combination of $T_n(x)$'s. However, I would like to use it as a part of an applied algorithm, and thus am concerned about the computational complexity of the formula; going through a monomial basis would be an overkill.

Is anyone aware of any such formula?