Some published papers and books give the impression that if you write down any infinite sequence of polynomials that follows a simple pattern, one will find that it's named after somebody and has an extensive literature. My question is whether the following is such a case: $ f_n(x) = nx + \binom n 3 x^3 + \binom n 5 x^5 + \cdots $ This has degree $n-1$ if $n$ is even and $n$ if $n$ is odd, so for each odd number there are two polynomials in the sequence with that degree.
A "known" polynomial sequence?
4
$\begingroup$
reference-request
polynomials
-
0@ColinMcQuillan : Maybe you weren't "notified" of the comments above, but herewith, you are. – 2012-12-23
1 Answers
5
Your formula has the following name: $f_n(x)=\frac{(1+x)^n-(1-x)^n}2$