| In this study, we combine some ideas from formal power series and symmetric functions to provide a uniform framework for proving congruences and identities. This setting permits us to uniformly explain relationships between Waring's Formulas, Newton's Identity, symmetric functions, and linear recurrence relations.; We have several different applications. In the first application, we use the cycle indicator Cn of the symmetric group and the Lagrange Inversion Theorem to derive various identities connecting several famous combinatorial sequences. In the second application, we discuss the relationship between the number of periodic points in a dynamical system, linear recurrence relations, and the power sum symmetric function in the characteristic roots of the recurrence relation. In the final application, we use our results to give explicit formulas for universal polynomials of universal lambda-rings. Moreover, we provide a connection of our work with ghost rings, necklace rings, and Witt vectors. |
