Fourth-order Linear Recurrence Research Articles

One kind of character sum modulo a prime p and its recurrence formula

The aim of this paper is to use an analytic method and the properties of the classical Gauss sums to research the computational problem of one kind of character sum of polynomials modulo an odd prime p and obtain several meaningful third- and fourth-order linear recurrence formulae for them.

On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums

Abstract The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime.

The hybrid power mean of the quartic Gauss sums and the two-term exponential sums

In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-order linear recurrence formula for it.

On the fourth-order linear recurrence formula related to classical Gauss sums

Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.

On the quartic Gauss sums and their recurrence property

The main purpose of this paper is, using the method of trigonometric sums and the properties of Gauss sums, to study the computational problem of one kind of congruence equation modulo an odd prime and give some interesting fourth-order linear recurrence formulas.

Pellans sequence and its diophantine triples

We introduce a novel fourth order linear recurrence sequence {Sn} using the two periodic binary recurrence. We call it ?pellans sequence? and then we solve the system ab+1=Sx, ac+1=Sy bc+1=Sz where a < b < c are positive integers. Therefore, we extend the order of recurrence sequence for this variant diophantine equations by means of pellans sequence.

A remark about the positivity problem of fourth order linear recurrence sequences

Consider a fourth order linear recurrence with integer coefficients whose characteristic polynomial has two distinct real and a complex conjugate pair of roots. A new proof showing that its positivity problem is decidable is given for the case where there is exactly one real root having the same absolute value as the two complex conjugate roots.

The positivity problem for fourth order linear recurrence sequences is decidable

The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.

Fourth Order Linearly Recurrent Wythoff Pairs

An examination of the sizable literature on Wythoff pairs, their generalizations, and Beatty sequences shows a considerable emphasis on connections with second order recurrence relations and quadratic irrationalities. However, it does not seem to have been previously noticed that a subsequence of the classical Wythoff pairs satisfies the irreducible fourth order linear recurrence $$C_{n + 4} = 10C_{n + 3} - 16C_{n + 2} + 5C_{n + 1} + C_n . $$ Thus, while Wythoff pairs are ordinarily associated with the roots of x2 − x − 1 = 0, there is a subset thereof that is associated with the roots of $$ x^4 - 10x^3 + 16x^2 - 5x - 1 = 0. $$

On a class of fourth order linear recurrence equations

This paper is concerned with sequences that satisfy a class of fourth order linear recurrence equations. Basic properties of such sequences are derived. In addition, we discuss the oscillatory and nonoscillatory behavior of such sequences.