Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

HOME ALL ARTICLES View

J. Korean Math. Soc. 2006; 43(1): 111-131

Printed January 1, 2006

Copyright © The Korean Mathematical Society.

$p$-adic $q$-higher-order hardy-type sums

Yilmaz Simsek

Akdeniz University

Abstract

The goal of this paper is to define $p$-adic Hardy sums and $p$-adic $q$-higher-order Hardy-type sums. By using these sums and $p$-adic $q$-higher-order Dedekind sums, we construct $p$-adic continuous functions for an odd prime. These functions contain $p$-adic $q$-analogue of higher-order Hardy-type sums. By using an invariant $p$-adic $q$-integral on $\mathbb{Z}% _{p}$, we give fundamental properties of these sums. We also establish relations between $p$-adic Hardy sums, Bernoulli functions, trigonometric functions and Lambert series.

Keywords: Dedekind sums, $p$-adic Dedekind sums, generalized Dedekind sums, Hardy sums, Bernoulli polynomials and functions, Lambert series $p$-adic $q$-higher order Dedekind sums, $p$-adic $q$-Bernoulli numbers

MSC numbers: Primary 11F20; Secondary11B68