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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.

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    $\begingroup$ A quick search (in English, French, German) on Google Books and Google Scholar in publications prior to 1900 returns a surprisingly small number of results, which seems to indicate that Chebyshev polynomials had been largely ignored by the mathematical world up to that point. Note that Chebyshev's name has been transliterated in about a dozen different ways, making literature searches more challenging. I believe I covered all of the common transliterations in my search. $\endgroup$
    – njuffa
    Dec 28, 2022 at 5:05
  • $\begingroup$ From P. Butzer and F. Jongmans, "P. L. Chebyshev (1821-1894): A guide to his life and work." Journal of approximation theory, Vol. 96, No. 1, Jan. 1999, pp. 111-138, it does not become clear how much uptake of Chebyshev's work there was in Western countries during his lifetime. He appears to have published mostly in French, so his results were (at least in principle) accessible to a Western audience. $\endgroup$
    – njuffa
    Dec 28, 2022 at 5:44
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    $\begingroup$ What is the meaning of notation $\circ$ ? Is it function composition ? $\endgroup$ Dec 29, 2022 at 21:32
  • $\begingroup$ Most families of (orthogonal) polynomials (Chebyshev, Laguerre, Hermite) were discovered in a small amount of time at the end of the 19th century ; and when I say orthogonal, I know that around 1900 this characterisation wasn't even understandable. $\endgroup$ Dec 29, 2022 at 21:40
  • $\begingroup$ I would not delve into Charles Émile Picard 's collected works, but the sequence $T_n(x)=\cos (n \arccos x)$ was the standard paradigm in his iteration sequences and their nesting features at around that time.However, I don't have hard references; merely inviting attention to those. $\endgroup$ Dec 30, 2022 at 21:45

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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.

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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.

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