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A polynomial is called self-reciprocal (or palindromic) if the sequence ofits coefficients is palindromic. In this paper we obtain improved error boundsfor the number of irreducible polynomials and self-reciprocal irreducible monicpolynomials with prescribed coefficients over a finite field. The improvedbounds imply that self-reciprocal irreducible monic polynomials with degree$2d$ always exist provided that the number of prescribed leading coefficientsis slightly less than $ d/2$.

Improved error bounds for the number of irreducible polynomials and  self-reciprocal irreducible monic polynomials with prescribed coefficients  over a finite field