Euler-Seidel matrices over Fp

A Euler–Seidel matrix is determined by an infinite sequence whose elements are given by recursion. The recurrence relations are investigated for numbers and polynomials such as hyperharmonics, Lucas numbers, and Euler and Genocchi polynomials. Linear recurring sequences in finite fields are employed, for instance, in coding theory and in several branches of electrical engineering. In this work, we define the period of a Euler–Seidel matrix over a field Fp with p elements, where p is a prime number. We give some results for the matrix whose initial sequence is {sr (n)}∞ n =0, where sr (n) = n ∑ k=0 (n, k)^r k, n ≥ 0 , and r is a fixed positive number. The numbers sr (n) play an important role in k=0 combinatorics and number theory. These numbers are known as Franel numbers for r = 3.