What was known about Chebyshev polynomials in 1900?

Question

Around 1900, was it widely known that the Chebyshev polynomial $T_n(X)$ satisfies the identity

$$ T_n(X) \circ \frac{X+X^{-1}}{2} = \frac{X^n+X^{-n}}2?$$

And also, would one expect top-notch algebraists of that era to be familiar with this (even though Chebyshev polynomials originated in analysis)?

I would have guessed that this identity was well known back then, since following Euler (if not earlier) it seems natural to rewrite the identity $T_n(\cos\theta)=\cos n\theta$ in terms of $z:=e^{i\theta}$, which yields the above identity.

Here is why I ask. In his PhD thesis of 1896, Dickson introduced a class of polynomials $D_n(X,a)$, which satisfy the identity

$$ D_n(X,a) \circ \Bigl(X+\frac{a}X\Bigr) = X^n + \Bigl(\frac{a}X\Bigr)^n,$$

and showed that they had remarkable properties over finite fields. What has always puzzled me is that Dickson didn't recognize the connection of these polynomials to Chebyshev polynomials, and especially that he didn't notice how similar his identity is to the Chebyshev polynomial identity mentioned above. Dickson didn't mention Chebyshev polynomials either in his thesis or in his 1901 book "Linear Groups". This seems especially surprising since Dickson was a first-rate mathematician, and also was extremely attentive to the previous math literature, for instance spending a decade writing his 1500-page history of the theory of numbers. So that's why I wonder which properties of Chebyshev polynomials would have been familiar to leading algebraists around 1900.

Answer 1

Chebyshev polynomials really appear in various connections since 16th century, see, for example https://arxiv.org/abs/2203.10955 But they were not named after Chebyshev until the early 20 century. So various authors just introduced them, and established those properties they need. For example, they occur in early 20 century holomorphic dynamics (Fatou, Julia, Lattes, Ritt), as exceptional cases, together with monomials and what is called now ``Lattes functions'', but none of these authors mentions Chebyshev, or refers to previous studies of these polynomials.

Answer 2

One might ask the same question about Lucas and his Lucas sequence $V_n := \alpha^n + \beta^n$ where $P:=\alpha+\beta$ and $Q:=\alpha\beta.$ The difference in the Dickson and Lucas cases seems to be that they are working with these numerical and polynomial sequences in the context of algebra and number theory and these sequences have two variables versus only one for the Chebyshev sequences. It also seems to me that there was no need to mention Chebyshev as this would add no actual useful information to the exposition of results. notice that the Lucas sequence Wikipedia article only has a very brief mention of Chebyshev polynomials.

The Wikipedia article Dickson polynomial states:

Since the Dickson polynomial $D_n(x,α)$ can be defined over rings with additional idempotents, $D_n(x,α)$ is often not related to a Chebyshev polynomial.