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Martin Himmel

Publications and source records attributed to Martin Himmel.

9 recordsLinked to original sources
arXiv2025

Homogeneity

The four types of homogeneity -- additive, multiplicative, exponential, and logarithmic -- are generalized as transformations describing how a function $f$ changes under scaling or shifting of its arguments. These generalized homogeneity functions capture different scaling behaviors and establish fundamental properties. Such properties include how homogeneity is preserved under function operations and how it determines the transformation behavior of related constructions like quotient functions. This framework extends the classical concept of homogeneity to a wider class of functional symmetries, providing a unified approach to analyzing scaling properties in various mathematical contexts.

math.GM↗
arXiv2021

Cauchy Pairs

The notion of pairable functions is introduced and some of its properties are developed. In this connection the famous Euler identity is interpreted as a property of certain pairable functions and finite cyclic groups.

math.GM↗
arXiv2021

Pendants to the Euler Beta function

Motivated by the integral representation of the Euler Beta function, we introduce its Cauchy siblings and investigate some of their properties. Two of these newly introduced functions happen to coincide with some classical means, such as the arithmetic or the logarithmic Cauchy one. Although the bivariable generalizations of Beta functions are obtained by elemantary integration, it seems difficult to obtain closed formulas for more than two variables. % The questions whether these Cauchy Beta functions belong to their respective class of Cauchy quotients is addressed and answered positively in the case of the Euler Beta function, but postponed to a future paper for all the other cases.

math.GM↗
arXiv2020

The logarithmic Cauchy quotient mean

Motivated by recent results on beta-type functions, a new family of means, which are of logarithmic Cauchy quotient type, are determined and characterized.

math.FA↗
arXiv2016

Homogeneous Beta-type functions

All beta-type functions, which are p-homogeneous, are determined. Applying this result, we show that a beta-type function is a homogeneous mean iff it is the harmonic one. A reformulation of a result due to Heuvers in terms of a Cauchy difference and the harmonic mean is given.

math.CA↗
arXiv2015

Directional convexity and characterizations of Beta and Gamma functions

The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \[ ϕ\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }ϕ\left( x\right) ,\ \ \ \ \ \ x>0, \] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function.

math.CA↗
arXiv2014

More convex functions by Artin`s method

First we recall the notion of conxity and log-convexity for real-valued. Then we generalize the trick used by Artin in his famous paper on the Gamma function to find log-convex solutions to the functional equations f(x+1)=g(x)f(x). This gives rise to understand a function by its representer.

math.CA↗