J. Korean Math. Soc. 2006; 43(1): 111-131
Printed January 1, 2006
Copyright © The Korean Mathematical Society.
Yilmaz Simsek
Akdeniz University
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
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