We develop a map Φ which sends a general even order self-reciprocal polynomial of degree 2n, denoted r_(2n)(x), to a polynomial of degree n denoted R_n(z), where z - x+x~(-1) and r_(2n)(x)=x~nR_n(z). The roots of R_n(z) determine those of r_(2n){x). We also prove that there are infinitely many even order self-reciprocal polynomials which can be reduced successively, n times, for each natural number n.
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