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 ReadingChebyshev’s other polynomials
There are two sequences of polynomials named after Chebyshev, and the first is so much more common that when authors…
Continue ReadingMore 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 ReadingAnalogy 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 ReadingProduct of Chebyshev polynomials
Chebyshev polynomials satisfy a lot of identities, much like trig functions do. This point will look briefly at just one…
Continue ReadingChebyshev approximation
In the previous post I mentioned that Remez algorithm computes the best polynomial approximation to a given function f as…
Continue ReadingGeneralization 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 ReadingGenerating 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 ReadingUniform 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 ReadingCounting 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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