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• # Artykuł - szczegóły

## Acta Arithmetica

2014 | 163 | 4 | 379-393

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.

379-393

wydano
2014

### Twórcy

autor
• Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, U.S.A.
autor
• Department of Mathematics, University of Kentucky, 719 Patterson Office Tower, Lexington, KY 40506, U.S.A.