Sylvester power and weighted sums on the Frobenius set in arithmetic progression

Alternate Sylvester sums on the Frobenius set

The alternate Sylvester sums are Tm(a, b) = ∑ n∈NR(−1)n, where a and b are coprime, positive integers, and NR is the Frobenius set associated with a and b. In this note, we study the generating functions, recurrences and explicit expressions of the alternate Sylvester sums. It can be found that the results are closely related to the Bernoulli polynomials, the Euler polynomials, and the (alterna...

Arithmetic-progression-weighted Subsequence Sums

Let G be an abelian group, let S be a sequence of terms s1, s2, . . . , sn ∈ G not all contained in a coset of a proper subgroup of G, and let W be a sequence of n consecutive integers. Let W ̄ S = {w1s1 + . . . + wnsn : wi a term of W, wi 6= wj for i 6= j}, which is a particular kind of weighted restricted sumset. We show that |W ̄S| ≥ min{|G| − 1, n}, that W ̄ S = G if n ≥ |G| + 1, and also c...

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

Weighted Frobenius-Perron and Koopman operators

Sylvester double sums and subresultants

Sylvester double sums versus subresultants given two polynomials A and B first notion symmetric expression of the roots of two polynomials second notion defined through the coefficients polynomials main result of the lecture these two notions are very closely related (idea due to Sylvester [S]) see details and complete proofs in [RS]. 1 Definitions and main result. A and B two finite families o...