Palindromic polynomial

A polynomial is palindromic, if the sequence of its coefficients are a palindrome.

Let $P(x)\; =\; sum\_\{i=0\}^n\; a\_ix^i$ be a polynomial of degree n, then P is palindromic if $a\_i\; =\; a\_\{n-i\}$ for i=0...n.

Similarly, P is called antipalindromic if $a\_i\; =\; -a\_\{n-i\}$ for i=0...n.

Examples

Some examples of palindromic polynomials are:

$(x+1)^2\; =\; x^2\; +\; 2x\; +\; 1$

$(x+1)^3\; =\; x^3\; +\; 3x^2\; +\; 3x\; +\; 1$

Generally, the expansion of $(x+1)^n$ is palindromic for all n (can see this from binomial expansion)

It also follows that if P is of even degree (so has odd number of terms in the polynomial), then it can only be antipalindromic when the 'middle' term is 0, i.e. $a\_i=-a\_i$, where $n=2i$.

ee also

* Reciprocal polynomial

External links

* [http://www.mathpages.com/home/kmath294.htm MathPages - The Fundamental Theorem for Palindromic Polynomials]