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Rademacher-Carlitz polynomials

Matthias BeckFlorian Kohl — 2014

Acta Arithmetica

We introduce and study the Rademacher-Carlitz polynomial R ( u , v , s , t , a , b ) : = k = s s + b - 1 u ( k a + t ) / b v k where a , b > 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 ( ( ( k a + 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...

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