Chebyshev polynomials of the second kind

There are two sequences of polynomials named after Chebyshev, and the first is so much more common that when authors say “Chebyshev polynomial” with no further qualification, they mean Chebyshev polynomials of the first kind. These are denoted with T_n, so they get Chebyshev’s initial [1]. The Chebyshev polynomials of the second kind are denoted U_n.

Chebyshev polynomials of the first kind are closely related to cosines. It would be nice to say that Chebyshev polynomials of the second kind are to sines what Chebyshev polynomials of the first kind are to cosines. That would be tidy, but it’s not true. There are relationships between the two kinds of Chebyshev polynomials, but they’re not that symmetric.

It is true that Chebyshev polynomials of the second kind satisfy a relation somewhat analogous to the relation

$\cos n\theta = T_n(\cos\theta)$

for his polynomials of the first kind, and it involves sines:

$\frac{\sin n\theta}{\sin \theta} = U_{n-1}(\cos\theta)$

We can prove this with the equation we’ve been discussing in several posts lately, so there is yet more juice to squeeze from this lemon.

Once again we start with the equation

$\cos n\theta + i\sin n\theta = \sum_{j=0}^n {n\choose j} i^j(\cos\theta)^{n-j} (\sin\theta)^j$

and take the complex part of both sides. The odd terms of the sum contribute to the imaginary part, so we can assume j = 2k + 1. We make the replacement

$(\sin\theta)^{2k+1} = (1 - \cos^2\theta)^k \sin\theta$

and so we’re left with a polynomial in cos θ, except for an extra factor of sin θ in every term.

This shows that sin nθ / sin θ, is a polynomial in cos θ, and in fact a polynomial of degree n − 1. Given the theorem that

$\frac{\sin n\theta}{\sin \theta} = U_{n-1}(\cos\theta)$

it follows that the polynomial in question must be U_n−1.

One thought on “Chebyshev’s other polynomials”

Michael Hannon

9 May 2020 at 17:37

A minor comment about the spelling: I conjecture that the initial ‘T’ stems from the time when French was the lingua franca. The guy’s last name in Russian is “Чебышёв”, and the initial letter is pronounced like the “ch” in “chop”. But “ch” at the start of a French word (as in Ma chérie amour, for instance) is pronounced like English “sh”. To recover the hard “ch” sound, the French would add an initial ‘T’. (BTW, the diaeresis over the second-to-last letter indicates that it is stressed and usually pronounced as “yo”, so the common, English pronunciations are all “wrong”.)

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One thought on “Chebyshev’s other polynomials”