Rademacher-Carlitz polynomials

• Autorzy: Matthias Beck , Florian Kohl
• Języki publikacji: EN

Abstrakty

EN

We introduce and study the Rademacher-Carlitz polynomial
$R(u,v,s,t,a,b) := ∑_{k=⌈s⌉}^{⌈s⌉+b-1} u^{⌊(ka+t)⌋}/b_{v^k}$
where $a,b ∈ ℤ_{>0}$, s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum
$r_t(a,b) := ∑_{k=0}^{b-1} (((ka+t)/b)) ((k/b))$,
which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz polynomials, we show how they are the only nontrivial ingredients of integer-point transforms
$σ(x,y) := ∑_{(j,k)∈𝓟 ∩ ℤ²} x^jy^k$
of any rational polyhedron 𝓟, and we derive the reciprocity theorem for Dedekind-Rademacher sums as a corollary which follows naturally from our setup.

Kategorie tematyczne

• 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
• 11F20: Dedekind eta function, Dedekind sums

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 163
• Numer: 4
• Strony: 379-393

Daty

wydano

2014

Twórcy

autor

Matthias Beck

• Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, U.S.A.

autor

Florian Kohl

• Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY 40506, U.S.A.

Identyfikatory

DOI

10.4064/aa163-4-6