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 Tn, so they get Chebyshev’s initial . The Chebyshev polynomials of the second kind are denoted Un.
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
for his polynomials of the first kind, and it involves sines:
Once again we start with the equation
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
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
it follows that the polynomial in question must be Un-1.
More special function posts
Chebyshev has been transliterated from the Russian as Chebysheff, Chebychov, Chebyshov, Tchebychev, Tchebycheff, Tschebyschev, Tschebyschef, Tschebyscheff, Chebychev, etc. It is conventional now to use “Chebyshev” as the name, at least in English, and to use “T” for the polynomials.