Generalized Cochrane sums and Cochrane-Hardy sums

In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

Degenerate and character Dedekind sums

In this paper, we study on two subjects. We first construct degenerate analogues of Dedekind sums in the sense of Apostol, Carlitz and Takács, and prove the corresponding reciprocity formulas. Secondly, we define generalized Dedekind character sums, which are explicit extensions of Berndt's definition, and prove the reciprocity laws. From the derived reciprocity laws, we obtain Berndt's reciprocity laws as special cases.     

An elliptic analogue of generalized Dedekind-Rademacher sums

In this paper we introduce an elliptic analogue of the generalized Dedekind-Rademacher sums which satisfy reciprocity laws. In these sums, Kronecker's double series play a role of elliptic Bernoulli functions. This paper gives an answer to the problem of S. Fukuhara and N. Yui concerning the elliptic Apostol-Dedekind sums. We also mention a relation between the generating function of Kronecker's double series and that of the (Debye) elliptic polylogarithms studied by A. Levin.     

On Wolstenholme's theorem and its converse

For any positive integer n, let . Wolstenholme proved that if p is a prime ?5, then . The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers wn, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers.     

Explicit evaluations of extended Euler sums

In this paper, we consider two types of extended Euler sums:

     

The fourth power of the Fermat quotient

Let p be an odd prime and qp(a)=(ap−1−1)/p be the Fermat quotient with base a, p?a. The main purpose of this paper is to investigate the fourth power problem of qp(2) and deduce an explicit formula represented by a linear combination of Mirimanoff polynomial values.     

Polynomials with two values

This paper investigates the minimal degree of polynomialsfR[x] that take exactly two values on a given range of integers {0,...n}. We show that thegap, defined asn-deg(f), isO(n ⁵⁴⁸). The maximal gap forn128 is 3. As an application, we obtain a bound on the Fourier degree of symmetric Boolean functions.     

Padé approximation and Apostol-Bernoulli and Apostol-Euler polynomials

Using the Padé approximation of the exponential function, we obtain recurrence relations between Apostol-Bernoulli and between Apostol-Euler polynomials. As applications, we derive some new lacunary recurrence relations for Bernoulli and Euler polynomials with gap of length 4 and lacunary relations for Bernoulli and Euler numbers with gap of length 6.     

On zeta function identities involving sums of squares and the zeta-theta correspondence

We prove two identities involving Dirichlet series, in the denominators of whose terms sums of two, three and four squares appear. These follow from two classical identities of Jacobi involving the four Jacobian Theta Functions θ₁(z;q), θ₂(z;q), θ₃(z;q) and θ₄(z;q), by the application of the Mellin transform. These results motivate the well-known correspondence between the set of the four Jacobian Theta Functions and the set of four classical zeta functions of which the Riemann Zeta Function is the third, and the Dirichlet Beta Function is the first.     

Evaluations of multiple Dirichlet L-values via symmetric functions

We explicitly evaluate a special type of multiple Dirichlet L-values at positive integers in two different ways: One approach involves using of symmetric functions, while the other involves using of a generating function of the values. Equating these two expressions, we derive several summation formulae involving the Bernoulli and Euler numbers. Moreover, values at non-positive integers, called central limit values, are also studied.     

On the ratios of the terms of second order linear recurrences

Research (partially) supported by Hungarian National Foundation for Scientific Research (OTKA) grant No. 273.     

On character sums of binary quadratic forms

We establish character sum bounds of the form

     

On distinct sums and distinct distances

The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Tóth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves that the number of distinct sums is at least n^d_s, where d_s=1/c_⌈s/2⌉ is defined in the paper and tends to e⁻¹ as s goes to infinity. Here e is the base of the natural logarithm. As an application we improve the Solymosi-Tóth bound on an old Erdős problem: we prove that n distinct points in the plane determine distinct distances, where ε>0 is arbitrary. Our bound also finds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.     

On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic

We employ the basic properties for the Hasse-Teichmüller derivatives to give simple proofs of known explicit formulae for Bernoulli numbers (of higher order) and then obtain some parallel results for their counterparts in positive characteristic.     

Categories with sums and right distributive tensor product

Models for parallel and concurrent processes lead quite naturally to the study of monoidal categories (Inform. Comput. 88 (2) (1990) 105). In particular a category Tree of trees, equipped with a non-symmetric tensor product, interpreted as a concatenation, seems to be very useful to represent (local) behavior of non-deterministic agents able to communicate (Enriched Categories for Local and Interaction Calculi, Lecture Notes in Computer Science, Vol. 283, Springer, Berlin, 1987, pp. 57-70). The category Tree is also provided with a coproduct (corresponding to choice between behaviors) and the tensor product is only partially distributive w.r.t. it, in order to preserve non-determinism. Such a category can be properly defined as the category of the (finite) symmetric categories on a free monoid, when this free monoid is considered as a 2-category. The monoidal structure is inherited from the concatenation in the monoid. In this paper we prove that for every alphabet A, Tree(A), the category of finite A-labeled trees is equivalent to the free category which is generated by A and enjoys the afore-mentioned properties. The related category Beh(A), corresponding to global behaviors is also proven to be equivalent to the free category which is generated by A and enjoys a smaller set of properties.     

A closed formula for the generating function of p-Bernoulli numbers

Abstract

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Bernoulli numbers and the generalized Bernoulli polynomials.     

On the properties of Newton-Euler pairs

If and are two sequences such that a₁=b₁ and , then we say that (a_n,b_n) is a Newton-Euler pair. In the paper, we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as and B_n, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and B_n is the nth Bernoulli number.