Two polynomials p(x) and q(x) are said to be permutable if p(q(x)) = q(p(x)) for all x. It’s not hard…

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## Negative space graph

Here is a plot of the first 30 Chebyshev polynomials. Notice the interesting patterns in the white space. Forman Acton…

Continue Reading## Chebyshev’s other polynomials

There are two sequences of polynomials named after Chebyshev, and the first is so much more common that when authors…

Continue Reading## More juice in the lemon

There’s more juice left in the lemon we’ve been squeezing lately. A few days ago I first brought up the…

Continue Reading## Analogy between Fibonacci and Chebyshev

Quick observation: I recently noticed that Chebyshev polynomials and Fibonacci numbers have analogous formulas. The nth Chebyshev polynomial satisfies for…

Continue Reading## Product of Chebyshev polynomials

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

Continue Reading## Chebyshev approximation

In the previous post I mentioned that Remez algorithm computes the best polynomial approximation to a given function f as…

Continue Reading## Generalization of power polynomials

A while back I wrote about the Mittag-Leffler function which is a sort of generalization of the exponential function. There…

Continue Reading## Generating De Bruijn cycles with primitive polynomials

A primitive polynomial of degree n with coefficients in GF(k), the finite field with k elements, has leading coefficient 1…

Continue Reading## Uniform approximation paradox

What I’m going to present here is not exactly a paradox, but I couldn’t think of a better way to…

Continue Reading## Counting irreducible polynomials over finite fields

You can construct a finite field of order pn for any prime p and positive integer n. The elements are…

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