Chebyshev polynomials satisfy a lot of identities, much like trig functions do. This point will look briefly at just one such identity.

Chebyshev polynomials *T*_{n} are defined for *n* = 0 and 1 by

*T*_{0}(*x*) = 1

*T*_{1}(*x*) = *x*

and for larger *n* using the recurrence relation

*T*_{n+1}(*x*) = 2*x**T*_{n}(*x*) − *T*_{n−1}(*x*)

This implies

*T*_{2}(*x*) = 2*x**T*_{1}(*x*) − *T*_{0}(*x*) = 2*x*^{2} − 1

*T*_{3}(*x*) = 2*x**T*_{2}(*x*) − *T*_{1}(*x*) = 4*x*^{3} − 3*x*

*T*_{4}(*x*) = 2*x**T*_{3}(*x*) − *T*_{2}(*x*) = 8*x*^{4} − 8*x*^{2} + 1

and so forth.

Now for the identity for this post. If *m* ≥ *n*, then

2 *T*_{m} *T*_{n} = *T*_{m+n} + *T*_{m–n}.

In other words, the product of the *m*th and *n*th Chebyshev polynomials is the average of the (*m* + *n*)th and (*m* – *n*)th Chebyshev polynomials. For example,

2 *T*_{3}(*x*) *T*_{1}(*x*) = 2 (4*x*^{3} − 3*x*) *x* = *T*_{4}(*x*) + *T*_{2}(*x*)

The identity above is not at all apparent from the recursive definition of Chebyshev polynomials, but it follows quickly from the fact that

*T*_{n}(cos θ) = cos *n*θ.

Proof: Let θ = arccos *x*. Then

2 T* _{m}*(

*x*)

*T*

_{n}(

*x*)

= 2 T

*(cos θ)*

_{m}*T*

_{n}(cos θ)

= 2 cos

*m*θ cos

*n*θ

= cos (

*m*+

*n*)θ + cos (

*m*−

*n*)θ

=

*T*

_{m+n}(cos θ) +

*T*

_{m−n}(cos θ)

=

*T*

_{m+n}(

*x*) +

*T*

_{m−n}(

*x*)

You might object that this only shows that the first and last line are equal for values of *x* that are cosines of some angle, i.e. values of x in [−1, 1]. But if two polynomials agree on an interval, they agree everywhere. In fact, you don’t need an entire interval. For polynomials of degree *m*+*n*, as above, it is enough that they agree on *m* + *n* + 1 points. (Along those lines, see Binomial coefficient trick.)

The close association between Chebyshev polynomials and cosines means you can often prove Chebyshev identities via trig identities as we did above.

Along those lines, we could have taken

*T*_{n}(cos θ) = cos *n*θ

as the definition of Chebyshev polynomials and then proved the recurrence relation above as a theorem, using trig identities in the proof.

Forman Acton suggested in this book Numerical Methods that Work that you should think of Chebyshev polynomials as “cosine curves with a somewhat disturbed horizontal scale.”

I think that the clearest approach to the Chebyshev polynomials is

just to combine De Moivre’s theorem and the binomial theorem.

Equating real parts gets you the usual Chebyshev polynomials,

and equating the imaginary parts and fiddling a bit gets you

Chebyshev polynomials of the second kind. It’s not too different

than just starting with cos nx = T_n(cos x), except that it is obvious

that T_n is a polynomial and the coefficients follow immediately.