The Bettin-Conrey reciprocity theorem and inflated Eulerian polynomials

This thesis is composed of two independent parts. The first part is motivated by a recent paper by Bettin and Conrey, introducing a family of cotangent sums that generalize the classical notion of Dedekind sum and share with it the property of satisfying a reciprocity law. We study particular instances of these arithmetic sums for which it is possible to obtain a simpler reciprocity. The second part focuses on the inflated s-Eulerian polynomial [...], introduced by Pensyl and Savage. We show that [...] is a polynomial for all positive integer sequences s and characterize those sequences s for which the sequence of nonzero coefficients of coincides with that of the polynomial [...]. In particular, we show that all nondecreasing sequences satisfy this condition.