A generalization of regular convolutions and Ramanujan sums

On a General Class of A-functions

“…The regular A-functions possess special combinatorial properties with respect to the corresponding A-convolution. Their main property is best exhibited in the following result, which can be seen, for example, as a consequence of Theorem 3 of [4]:…”

“…Certain classes of A-functions have been the subject of many manuscripts devoted to exploring their properties. The regular A-functions of Narkiewicz [23] in particular have been heavily investigated and generalized (see [4,14,18,22,24,29,30,32,34]).…”

“…We further show how every complete A-function may be associated to a perfect A-function. Examples abound throughout this section, including a special focus on the A-function G from [4].…”

“…In keeping with the notation of [4], for a simple and reflexive A-function A, we define µ A to be the left inverse of u under the A-convolution * A . Note that this generalizes the definition of the usual Möbius function to all applicable A.…”

“…Regular A-functions defined in this manner are difficult to work with. Fortunately, Narkiewicz and others [4,23,30] have developed more workable definitions, which we state as follows:…”