Editing Reciprocal polynomial (section)

{{Short description|Polynomial with reversed root positions}}
{{More citations needed|date=April 2021}}
In [[algebra]], given a [[polynomial]]
:<math>p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n,</math>
with [[coefficient]]s from an arbitrary [[Field (mathematics)|field]], its '''reciprocal polynomial''' or '''reflected polynomial''',<ref name="concrete">*{{cite book | last1 = Graham | first1 = Ronald |last2=Knuth|first2=Donald E.|last3=Patashnik|first3=Oren | title = Concrete mathematics : a foundation for computer science | edition=Second | publisher = Addison-Wesley | location = Reading, Mass | year = 1994 | isbn = 978-0201558029 | page= 340}}</ref><ref name="Aigner">{{cite book | last = Aigner | first = Martin | title = A course in enumeration | publisher = Springer | location = Berlin New York | year = 2007 | isbn = 978-3540390329 | page = 94 }}</ref> denoted by {{math|''p''<sup>∗</sup>}} or {{math|''p''<sup>R</sup>}},<ref name="Aigner"/><ref name="concrete"/> is the polynomial<ref>{{harvnb|Roman|1995|loc=pg.37}}</ref>

:<math>p^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n p(x^{-1}).</math>

That is, the coefficients of {{math|''p''<sup>∗</sup>}} are the coefficients of {{math|''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 number]]s, when

:<math>p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n,</math>

the '''conjugate reciprocal polynomial''', denoted {{math|''p''<sup>†</sup>}}, is defined by,

:<math>p^{\dagger}(z) = \overline{a_n} + \overline{a_{n-1}}z + \cdots + \overline{a_0}z^n = z^n\overline{p(\bar{z}^{-1})},</math>

where <math>\overline{a_i}</math> denotes the [[complex conjugate]] of <math>a_i</math>, and is also called the reciprocal polynomial when no confusion can arise.

A polynomial {{math|''p''}} is called '''self-reciprocal''' or '''palindromic''' if {{math|1=''p''(''x'') = ''p''<sup>∗</sup>(''x'')}}.
The coefficients of a self-reciprocal polynomial satisfy {{math|1=''a''<sub>''i''</sub> = ''a''<sub>''n''−''i''</sub>}} for all {{math|''i''}}.