Year: 2025
Type: article
Abstract: Let S be a semigroup, $$\sigma :S \rightarrow S$$ σ : S → S a surjective homomorphism and $$\mathbb {F}$$ F a field of characteristic different from 2. We study solutions $$f,g :S \rightarrow \mathbb {F}$$ f , g : S → F of the functional equation $$\begin{aligned} f(x\sigma (y)) = f(x)g(y) + \beta g(x)f(y) + \gamma f(x)f(y), \ x,y \in S, \end{aligned}$$ f ( x σ ( y ) ) = f ( x ) g ( y ) + β g ( x ) f ( y ) + γ f ( x ) f ( y ) , x , y ∈ S , in which $$\beta \in \mathbb {F}^*$$ β ∈ F ∗ and $$\gamma \in \mathbb {F}$$ γ ∈ F are constants. The functional equation generalizes the sine addition and subtraction laws. We show that each solution ( f , g ) such that f and g are linearly independent, satisfies the special cosine-sine functional equation $$\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) + cf(x)f(y), \ x,y \in S, \end{aligned}$$ f ( x y ) = f ( x ) g ( y ) + g ( x ) f ( y ) + c f ( x ) f ( y ) , x
Source: Aequationes Mathematicae
Author: Henrik Stetkær
Institution: Aarhus University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
Funder: Aarhus Universitet
OpenAlex ID: W4411206523