A brief account of our recent works dealing with the problem of characterizing hyperbolic functions through the use of functional equations with side conditions is reported. Specifically, the fact that each of the n functions having a zero sum over n points lying in a hyperplane is necessarily a linear function, is generalized and the results so obtained are applied to characterize hyperbolic functions.

/pdf/characterizing-hyperbolic-functions-over-a-hyperplane-40ljfy4w4f.pdf

Characterizing Hyperbolic Functions over a Hyperplane

In this paper, we consider $k$-free numbers over Beatty sequences. New results are given.

On $k$-free numbers over Beatty sequences

Consecutive generalized r-free integers in Beatty sequences

An investigation about a generalized unitary convolution of two arithmetic functions, its basic properties and its connections to generalized Ramanujan sums is carried out with applications. The results so obtained complement our earlier work in the Dirichlet convolution setting. 

Generalized Unitary Convolution, Ramanujan Sums and Applications

Let l be a given nonzero integer. The authors give an explicit characterization of the positive integer k that makes the Diophantine equation x2 − kxy + y2 +lx = 0 have infinitely many positive integer solutions (x, y).

On the Diophantine Equation x2-kxy+y2+lx=0.

On r-free integers in Beatty sequences

Abstract We consider the Diophantine equation x2-kxy+ky2+ ly = 0 for l = 2n and determine for which values of the odd integer k, it has a positive integer solution x and y.

/pdf/on-the-diophantine-equation-x2-kxy-ky2-ly-0-l-2n-hp19e58w76.pdf

On the Diophantine Equation x2 – kxy + ky2 + ly = 0, l = 2n

Two functional equations exhibiting functions with constant sums over points
 lying in a hyperplane are solved, and the results are applied to
 characterize major trigonometric and hyperbolic functions.

/pdf/functions-with-constant-sums-over-a-hyperplane-and-4mvojpyajj.pdf

Functions with constant sums over a hyperplane and applications

Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1, . . . , k − 1}2 satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural restrictions. Some of these formulae are obtained as consequences of finite Fourier series expansions of the greatest common divisor function, whereas some are shown to be closely connected with the notion of unitary divisors.

Sukrawan Mavecha

Papers

On r-free integers in Beatty sequences

On the Diophantine Equation x2 – kxy + ky2 + ly = 0, l = 2n

Functions with constant sums over a hyperplane and applications

Integer factorization and finite Fourier series expansion

Consecutive generalized r-free integers in Beatty sequences