In algebra, given a polynomial
p ( x ) = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n , {\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by p∗ or pR, is the polynomial
p ∗ ( x ) = a n + a n − 1 x + ⋯ + a 0 x n = x n p ( x − 1 ) . {\displaystyle p^{*}(x)=a_{n}+a_{n-1}x+\cdots +a_{0}x^{n}=x^{n}p(x^{-1}).}
That is, the coefficients of p∗ are the coefficients of p in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when
p ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n , {\displaystyle p(z)=a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n},}
the conjugate reciprocal polynomial, denoted p†, is defined by,
p † ( z ) = a n ¯ + a n − 1 ¯ z + ⋯ + a 0 ¯ z n = z n p ( z ¯ − 1 ) ¯ , {\displaystyle p^{\dagger }(z)={\overline {a_{n}}}+{\overline {a_{n-1}}}z+\cdots +{\overline {a_{0}}}z^{n}=z^{n}{\overline {p({\bar {z}}^{-1})}},}
where a i ¯ {\displaystyle {\overline {a_{i}}}} denotes the complex conjugate of a i {\displaystyle a_{i}} , and is also called the reciprocal polynomial when no confusion can arise. A polynomial p is called self-reciprocal or palindromic if p(x) = p∗(x). The coefficients of a self-reciprocal polynomial satisfy ai = an−i for all i.
Properties Reciprocal polynomials have several connections with their original polynomials, including:
deg p = deg p∗ if a 0 {\displaystyle a_{0}} is not 0. p(x) = xnp∗(x−1). α is a root of a polynomial p if and only if α−1 is a root of p∗ or if α and p∗ is of lower degree than p. If x ∤ p(x) then p is irreducible if and only if p∗ is irreducible. p is primitive if and only if p∗ is primitive. Other properties of reciprocal polynomials may be obtained, for instance:
A self-reciprocal polynomial of odd degree is divisible by x+1, hence is not irreducible if its degree is > 1.
Palindromic and antipalindromic polynomials A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
P ( x ) = ∑ i = 0 n a i x i {\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}
is a polynomial of degree n, then P is palindromic if ai = an−i for i = 0, 1, ..., n. Similarly, a polynomial P of degree n is called antipalindromic if ai = −an−i for i = 0, 1, ..., n. That is, a polynomial P is antipalindromic if P(x) = –P∗(x).
Examples From the properties of the binomial coefficients, it follows that the polynomials P(x) = (x + 1)n are palindromic for all positive integers n, while the polynomials Q(x) = (x – 1)n are palindromic when n is even and antipalindromic when n is odd. Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
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