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Reducibility and irreducibility of Stern ( 0 , 1 ) -polynomials

Karl Dilcher, Larry Ericksen (2014)

Communications in Mathematics

The classical Stern sequence was extended by K.B. Stolarsky and the first author to the Stern polynomials a ( n ; x ) defined by a ( 0 ; x ) = 0 , a ( 1 ; x ) = 1 , a ( 2 n ; x ) = a ( n ; x 2 ) , and a ( 2 n + 1 ; x ) = x a ( n ; x 2 ) + a ( n + 1 ; x 2 ) ; these polynomials are Newman polynomials, i.e., they have only 0 and 1 as coefficients. In this paper we prove numerous reducibility and irreducibility properties of these polynomials, and we show that cyclotomic polynomials play an important role as factors. We also prove several related results, such as the fact that a ( n ; x ) can only have simple zeros, and we state a...

Remarks on Steinhaus’ property and ratio sets of sets of positive integers

Tibor Šalát (2000)

Czechoslovak Mathematical Journal

This paper is closely related to an earlier paper of the author and W. Narkiewicz (cf. [7]) and to some papers concerning ratio sets of positive integers (cf. [4], [5], [12], [13], [14]). The paper contains some new results completing results of the mentioned papers. Among other things a characterization of the Steinhaus property of sets of positive integers is given here by using the concept of ratio sets of positive integers.

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