Arthur Shagulian: Generating Functions and Applications

A generating function is a formal power series that contains information about a sequence of numbers. Applications of generating functions are many. They are used in a broad field of study and are powerful tools used in different type of computational problems. In this paper we shall be mainly concerned about two applications: i) Exact Covering Sequence (ECS): referring to a finite sequence of residue classes in which every non-negative integer is covered by one and only one congruent; by using generating functions, cyclotomic polynomials and Möbius inversion formula, we shall show whether any given set of residue classes can be an ECS. ii) Calculation of the number of Square Roots of a given Permutation: Here we are going to use cyclic index of the symmetric group to create an exponential generating function which shall provide us with the number of permutations of a given set with possible square roots.