Abstract
AbstractIn this paper we consider a generalization of d’Alembert’s equation and Wilson’s equation on commutative semigroups using only the semigroup operation, ie. we consider the functional equation $$\begin{aligned}&h(x+2y)+h(x)=2f(y)h(x+y),\ x,y\in S, \end{aligned}$$
h
(
x
+
2
y
)
+
h
(
x
)
=
2
f
(
y
)
h
(
x
+
y
)
,
x
,
y
∈
S
,
where $$f,h:S\rightarrow \mathbb {K}$$
f
,
h
:
S
→
K
, $$(S,+)$$
(
S
,
+
)
is a commutative semigroup, $$\mathbb {K}$$
K
is a quadratically closed field, $$\text {char}\,\mathbb {K}\ne 2$$
char
K
≠
2
.
Funder
University of Silesia in Katowice
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,General Mathematics
Reference22 articles.
1. Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)
2. Aczél, J., Chung, J.K., Ng, C.T.: Symmetric second differences in product form on groups. Top. Math. Anal. Ser. Pure Math. 11, 1–12 (1989)
3. Badora, R.: On the d’Alembert type functional equation in Hilbert algebras. Funkcial. Ekvac. 43, 405–418 (2000)
4. Baker, J.A.: D’Alembert’s functional equation in Banach algebras. Acta Sci. Math. (Szeged) 32, 225–234 (1971)
5. Benson, C., Jenkins, J., Ratcliff, G.: On Gelfand pairs associated with solvable Lie groups. Trans. Am. Math. Soc. 321, 85–116 (1990)