Vol. 75, No. 2, 1978

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Some theorems on generalized Dedekind-Rademacher sums

L. Carlitz

Vol. 75 (1978), No. 2, 347–358
Abstract

Radamacher has defined a generalized Dedekind sum

s(h,k;x,y) =  ∑    ((h a+-y-+ x))((a+-y))
a (mod k)    k           k

and proved a reciprocity theorem for this sum that generalizes the well known result for s(h,k). In the present paper we define

ϕr,s(h,k;x,y) = a (mod k)Br(h(a+-y-
k) + x)Bs(a-+-y
k),
ψr.s(h,k;x,y) = j=0r(1)rj(r)
jhrjϕ j,r+sj(h,k;x,y),
where Bn(x) is the Bernoulli function, and show that
      s                       r
(s + 1)k ψr+1,s(h,k;x,y)− (r+ 1)h ψs+1,r(k,h;y,x)-
= (s +1)kBr+1(x)Bs(y)− (r+ 1)hBr(x)Bs+1(y)  ((h,k) = 1).

We also prove the polynomial reciprocity theorem

      k∑−1 h−[(ha+z)∕k] a        h∑−1 k−[(kb+z)∕h] b   h   k
(1− v)   u          v − (1− u)   v         u  = u − v   ((h,k ) = 1)
a=0                     b=0

as well as some related results.

Mathematical Subject Classification
Primary: 10A20, 10A20
Secondary: 10A15
Milestones
Received: 13 September 1976
Published: 1 April 1978
Authors
L. Carlitz