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We compute bilinear integrals involving Macdonald and Gegenbauer functions. These integrals are convergent only for a limited range of parameters. However, when one uses generalized integrals they can be computed essentially without…

Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic…

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the…

Using known entropic and information inequalities new inequalities for some classical polynomials are obtained. Examples of Jacobi and Legendre polynomials are considered.

We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical…

We study the elliptic maximal functions defined by averages over ellipses and rotated ellipses which are multi-parametric variants of the circular maximal function. We prove that those maximal functions are bounded on $L^p$ for some $p\neq…

A classical problem in Complex Dynamics on hyperbolic domains is to characterize the hyperbolic step of parabolic functions. This topic has been studied by several authors, leading to different results and providing characterizations that…

A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…

We define and study hyperbolic extensions.

In 2006, in a paper published in Compositio titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer derived bounds for the canonical Green's function and the hyperbolic Green's function defined on a compact hyperbolic…

In this paper, we present series representations of the remainders in the expansions for certain trigonometric and hyperbolic functions. By using the obtained results, we establish some inequalities for trigonometric and hyperbolic…

This paper deals with generalized elliptic integrals and generalized modular functions. Several new inequalities are given for these and related functions.

We extend the classical Copson's inequalities so that the values of parameters involved go beyond what is currently known.

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on…

This paper deals with some inequalities for trigonometric and hyperbolic functions such as the Jordan inequality and its generalizations. In particular, lower and upper bounds for functions such as (sin x)/x and x/(sinh x) are proved.

Motivated by the work of J. S\'andor [19], in this paper we establish a new Wilker type and Huygens type inequalities involving the trigonometric and hyperbolic functions. Moreover, in terms of hyperbolic functions, the upper and lower…

The parabolic trigonometric functions have recently been introduced as an intermediate step between circular and hyperbolic functions. They have been shown to be expressible in terms of irrational functions, linked to the solution of third…

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize…

In the paper, the authors prove that the generalized sine function $\sin_{p,q}(x)$ and the generalized hyperbolic sine function $\sinh_{p,q}(x)$ are geometrically concave and geometrically convex, respectively. Consequently, the authors…

Two distinct systems of commutative complex numbers in n dimensions are described, of polar and planar types. Exponential forms of n-complex numbers are given in each case, which depend on geometric variables. Azimuthal angles, which are…