scholarly journals p-Adic q-Twisted Dedekind-Type Sums

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1756
Author(s):  
Abdelmejid Bayad ◽  
Yilmaz Simsek
Keyword(s):  
Bernoulli Polynomials ◽  
Dedekind Sums ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Finite Sums ◽  
Symmetry Relations

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral and the Teichmüller character representations of the Bernoulli polynomials, we give reciprocity law of these sums. These sums and their reciprocity law generalized some of the classical p-adic Dedekind sums and their reciprocity law. It is to be noted that the Dedekind reciprocity laws, is a fine study of the existing symmetry relations between the finite sums, considered in our study, and their symmetries through permutations of initial parameters.

scholarly journals P-adic interpolation of dedekind sums

1988 ◽  
Vol 37 (2) ◽  
pp. 293-301 ◽  
Author(s):  
C. Snyder
Keyword(s):  
Positive Integer ◽  
Measure Theory ◽  
Explicit Representation ◽  
Dedekind Sums ◽  
Reciprocity Laws ◽  
Reciprocity Law ◽  
Dirichlet Characters

In this article we give an explicit representation of p-adic Dedekind sums and their reciprocity laws by using p-adic measure theory. We then study the consequences of the p-adic reciprocity law for particular positive integer values in which case we can recover a reciprocity law for Dedekind sums attached to particular Dirichlet characters. This gives a proof different from that of Nagasaka.

scholarly journals Generalizations of Dedekind sums and their reciprocity laws

2003 ◽  
Vol 106 (4) ◽  
pp. 355-378 ◽  
Author(s):  
Yumiko Nagasaka ◽  
Kaori Ota ◽  
Chizuru Sekine
Keyword(s):  
Dedekind Sums ◽  
Reciprocity Laws

scholarly journals Franel integrals of order four

1996 ◽  
Vol 60 (2) ◽  
pp. 192-203 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Lattice Point ◽  
Elementary Proof ◽  
Unit Interval ◽  
Dedekind Sums ◽  
Dedekind Sum ◽  
Elementary Method ◽  
The Third ◽  
Reciprocity Law ◽  
Order Four

AbstractLet ((x)) =x−⌊x⌋−1/2 be the swatooth function. Ifa, b, cand e are positive integeral, then the integral or ((ax)) ((bx)) ((cx)) ((ex)) over the unit interval involves Apolstol's generalized Dedekind sums. By expressing this integral as a lattice-point sum we obtain an elementary method for its evaluation. We also give an elementary proof of the reciprocity law for the third generalized Dedekind sum.

scholarly journals Some remarks on regular integers modulo n

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 687-701
Author(s):  
Brăduţ Apostol ◽  
László Tóth
Keyword(s):  
Gamma Function ◽  
Bernoulli Polynomials ◽  
Weak Order ◽  
Cyclotomic Polynomials ◽  
Power Sums ◽  
Multidimensional Generalization ◽  
Maximal Orders ◽  
Finite Sums

An integer k is called regular (mod n) if there exists an integer x such that k2x ? k (mod n). This holds true if and only if k possesses a weak order (mod n), i.e., there is an integer m ? 1 such that km+1 ? k (mod n). Let ?(n) denote the number of regular integers (mod n) in the set {1,2,...,n}. This is an analogue of Euler?s ? function. We introduce the multidimensional generalization of ?, which is the analogue of Jordan?s function. We establish identities for the power sums of regular integers (mod n) and for some other finite sums and products over regular integers (mod n), involving the Bernoulli polynomials, the Gamma function and the cyclotomic polynomials, among others. We also deduce an analogue of Menon?s identity and investigate the maximal orders of certain related functions.

Multiple Dedekind–Rademacher sums in function fields

2014 ◽  
Vol 10 (05) ◽  
pp. 1291-1307 ◽  
Author(s):  
Abdelmejid Bayad ◽  
Yoshinori Hamahata
Keyword(s):  
Function Fields ◽  
Dedekind Sums ◽  
Reciprocity Law ◽  
Higher Dimensional

In the previous paper, we introduced the higher-dimensional Dedekind sums in function fields, and established the reciprocity law. In this paper, we generalize our higher-dimensional Dedekind sums and establish the reciprocity law and the Petersson–Knopp identity.

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