Abstract
We prove the polynomial analogues of some Liouville identities from elementary number theory. Consequently several sums defined over the finite fields Fq[t] are evaluated by combining the results obtained and some of the results from sums of reciprocals of polynomials over Fq[t].
Highlights
Introduction and BackgroundLet p be a prime number, q = pe for some positive integer e and Fq the finite field with q elements
For the ring Z, the units are ±1 and every nonzero integer is a multiple by a unit of a positive integer
Analogues of the little theorem of Fermat and Euler, Wilsons theorem, quadratic reciprocity, the prime number theorem, and Dirichlet’s theorem on primes in arithmetic progression and many more well-known theorems from elementary number theory have been proven true for Fq[t], see [1]
Summary
Joseph Liouville introduced a powerful new method into elementary number theory that allowed him to get many interesting identities His approach was used to solve well-known problems such as sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums, and many others. L≡b(mod m) where [x] represents the integer n satisfying n ≤ x < n + 1 He proved many other results related to Theorem 1. We evaluate the average of some functions defined on the set of all monic polynomials over Fq. Throughout this paper, we let A = Fq[t]. We give a closed formula for some sums of reciprocals of polynomials over Fq[t] These results will be used in the subsequent sections to help prove some polynomial identities in Fq[t].
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