Abstract
Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1,?, mn,?, and on a sequence of complex numbers z1,?, zn, ? of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived.
Highlights
Recall that the nth cyclotomic polynomial Φn is defined by (1)Φn(z) = (z − ζ), ζ n =1 where ζ are the primitive roots of order n of the unity
As particular cases we obtain the extended cyclotomic, the extended polygonal-type, and the multinomial polynomials. For these classes of polynomials we establish integral formulae for the coefficients (Section 3), which are useful in the study of the asymptotic behaviour of the coefficients
By Theorem 5 (2) we get an integral formula for the coefficients of m m1,···,ms
Summary
As particular cases we obtain the extended cyclotomic, the extended polygonal-type, and the multinomial polynomials For these classes of polynomials we establish integral formulae for the coefficients (Section 3), which are useful in the study of the asymptotic behaviour of the coefficients. In the last section we generate certain sequences defined by the number of non-zero coefficients or the maximum coefficient, and the number of irreducible factors over integers of these polynomials. In this process we obtain some novel sequences, not currently indexed in the Online Encyclopedia of Integer Sequences (OEIS), or new meanings to existing entries
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