In this paper, we utilize the \(\sigma\) category to introduce \(EKFN\)-modules, which extend the concept of the \(EKFN\)-ring. After presenting some properties, we demonstrate, under certain hypotheses, that if \(M\) is an \(EKFN\)-module, then the following equivalences hold: the class of uniserial modules coincides with the class of \(cu\)-uniserial modules; \(EKFN\)-modules correspond to the class of locally noetherian modules; and the class of \(CD\)-modules is a subset of the \(EKFN\)-modules.

Let \(G\) be a finite solvable group and \(\Delta\) be the subset of \(\Upsilon \times \Upsilon\), where \(\Upsilon\) is the set of all pairs of size two commuting elements in \(G\). If \(G\) operates on a transitive \(G\) – space by the action \((\upsilon_{1},\upsilon_{2})^{g}=(\upsilon_{1}^{g},\upsilon_{2}^{g})\); \(\upsilon_{1},\upsilon_{2} \in \Upsilon\) and \(g \in G\), then orbits of \(G\) are called orbitals. The subset \(\Delta_{o}=\{(\upsilon,\upsilon);\upsilon \in \Upsilon, (\upsilon,\upsilon) \in \Upsilon \times \Upsilon\}\) represents \(G’s\) diagonal orbital.
The orbital regular graph is a graph on which \(G\) acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.

By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.