Arithmetical and geometrical means of generalized logarithmic and exponential functions: generalized sum and product operators

TL;DR

This paper introduces generalized logarithmic and exponential functions with algebraic operators, providing analytical expressions for their iterative applications and exploring their potential applications.

Contribution

It presents new algebraic operators for generalized logarithmic and exponential functions, along with analytical formulas for their repeated applications.

Findings

Derived explicit formulas for successive applications of generalized sum and product operators.

Demonstrated potential applications of the generalized formalism.

Extended classical logarithmic and exponential functions through one-parameter generalizations.

Abstract

One-parameter generalizations of the logarithmic and exponential functions have been obtained as well as algebraic operators to retrieve extensivity. Analytical expressions for the successive applications of the sum or product operators on several values of a variable are obtained here. Applications of the above formalism are considered.