On a generalized convolution of incidence functions Original Research Article

Let (P,⩽) be a locally finite partially ordered set. Let K(x,y;z) be a function of x,y,z∈P. We define the K-convolution of incidence functions f and g as(f★g)(x,y)=∑x⩽z⩽yf(x,z)g(z,y)K(x,y;z).We define two transformations on the set of incidence functions, which serve as logarithm and exponential operators under the K-convolution, give their basic properties and apply them in finding solutions for the functional equations f(r)=g, f(r)=fg and f★g=h in f, where f(r) denotes the rth iterate of f with respect to the K-convolution. These results cast some known results on arithmetical functions in the poset-theoretic framework.