What does this research mean for the field?

All solutions to the functional equation f(f(x+y))=f(x+y)+f(x)f(y) can be determined and are connected with ideals in a group algebra. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This paper aims to identify all solutions to the functional equation involving transcendental numbers.

May 16, 2026Open Access

On the solutions of a functional equation attaining transcendental numbers only

JSJohannes SchoißengeierUniversity of Vienna

Key Points

This paper aims to identify all solutions to the functional equation involving transcendental numbers.
Examine the functional equation f(f(x+y))=f(x+y)+f(x)f(y) for real numbers x and y.
Establish conditions under which non-trivial solutions exist, particularly focusing on f(0)=c with c being transcendental.
Discuss the odd properties of solutions and their relationship with ideals in group algebra.
Identified conditions for non-trivial solutions where f(0)=c, specifically when c is transcendental and greater than -1.
Demonstrated that under these conditions, all values of f are transcendental.
Explored the odd properties of solutions and their connections to group algebra ideals.

Abstract

Abstract We investigate the functional equation f (f (x+y) ) =f (x+y) +f (x) f (y), \ x, y R. f (f (x + y) ) = f (x + y) + f (x) f (y), x, y ∈ R. Given a real number c 0, c ≠ 0, it was shown in 5 that there is a non-trivial solution f with f (0) =c f (0) = c iff c is transcendental and >-1 > - 1. In that case all values of f are transcendental. In this paper we find all solutions of the functional equation, discuss their odd properties and the connection with ideals in a group algebra.

AI से पूछें

Bookmark

View Full Paper