# Where does the Chebyshev polynomial notation come from?

The $k$th Chebyshev polynomial is denoted by $T_k$ where

$T_k(x) = \cos(k\cos^{-1}(x))$

I was wondering where this notation came from. It has been suggested that it comes from Tschebyscheff (the Russian name for Chebyshev) but does anyone know the first use of this notation or verify this is the reason?

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The cyrillic way of writing Chebyshev starts with a letter looking a bit like a 4, not with a T. In German transliterations, however, a T was used for the first letter (Tschebyscheff, Tchebyshew, and other variations). – Franz Lemmermeyer Oct 10 '10 at 7:11
As Franz explained, there are at least two reasons, why the original paper is not the right place to look for the letter 'T'. Firstly, why would Chebyshev (sic) call a polynomial after himself (assuming that the modern 'T' refers to his name) and secondly, in Russian the name is Чебышев and starts with a 'Ч'. It must have been a German or a French who introduced the letter 'T'. – Alex B. Oct 10 '10 at 8:59
Reminds me of that anecdote about Besicovitch... – J. M. Oct 10 '10 at 9:29
@Alex: Exactly! Do you have any suggestions on where to start looking? – alext87 Oct 10 '10 at 10:59

Great Soviet mathematician N.I. Akhiezer mentions in his survey article "Чебышевское направление в теории функций" ("Function Theory According to Chebyshev") that the notation $$T_n(x)=\frac{1}{2^{n-1}} \cos{(n\arccos x)}$$ was first introduced by S. Bernstein.

I think that the first published paper on the Chebyshev polynomials by Bernstein was "О наилучшем приближении непрерывных функций посредством полиномов многочленов данной степени" which appeared in "Сообщения Харьковского математического общества", series 2, vol. 13 (1912), pp. 49-194. The paper is in Russian as you may guess.

In his paper, Bernstein refers to the Chebyshev polynomials as trigonometric polynomials which probably might explain the letter T in the notation.

English translation of Akhiezer's survey article is contained in Mathematics of the 19th Century edited by A.N. Kolmogorov.

Edit added. I don't know if there is an English translation of the original paper by Bernstein. This source refers to the paper as "The optimum approximation to continuous functions by polynomials of a given power", Reports of the Kharkov Mathematical Society, Second Series, 1912, 13, #2-3. The original paper can be also found in volume 1 of S.N. Bernstein Collected Works (С.Н. Бернштейн, Собрание сочинений (Том 1. Конструктивная теория функций [1905-1930]), Москва, 1952).

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Thank you! That's great! – alext87 Oct 11 '10 at 7:03
You're welcome. – Andrey Rekalo Oct 11 '10 at 18:06
Being a bit nit-picky: the title of Bernstein's paper is "О наилучшем приближении непрерывных функций посредством многочленов данной степени" - although polinom and mnogochlen has the same meaning, the original title contains the latter, according to reference bases. – Harun Šiljak Oct 11 '10 at 18:15
@H. M. Šiljak: Good point! Thanks. – Andrey Rekalo Oct 11 '10 at 18:19