Identities Involving the Fourth-Order Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

Yuankui Ma and Wenpeng Zhang (see [17]) acquired a different expression about the summation by introducing a new second order non-linear recursive sequence.
In [18], Taekyun Kim and others studied the properties of Fibonacci numbers by introducing the convolved Fibonacci numbers p n (x), which are given by the generating function The authors gave a new formula for calculating p n (x) by the elementary and combinatorial methods, and obtained some new and explicit identities of the convolved Fibonacci numbers, including the relationship between p n (x) and the combination sums about Fibonacci numbers.
In this paper, we consider the Tetranacci numbers H n (see [19]), which are defined by the fourth-order linear recurrence relation The Tetranacci numbers can be extended to negative index n arising from the rearranged recurrence relation which yields the sequence of "nega-Tetranacci" numbers, The generating function of the Tetranacci sequences H n is given by Tetranacci numbers have important applications in combinatorial counting and graph theory, W. Marcellus E (see [20,21]) studied the arithmetical properties of H n , Rusen Li (see [22]) obtained some convolution identities for H n . Moreover, the summation calculation for different sequences is one of the hot topics in number theory, and many scholars have obtained a series of interesting results (see [23,24]). Therefore, it is very meaningful to further study the properties of the Tetranacci sequences. Inspired by the above references, for a real number x ∈ R, we can define a new function H n (x), which is given by The main purpose of this paper is to study the relationship between H n (x) and H n , and to prove some computational formulas of the fourth-order recurrence sequence by applying the elementary method and the symmetry properties of the summation processes. That is, we shall prove the following: Theorem 1. For a real number x ∈ R and any integer n ≥ 0, we have ∑ a+b+c+d=n denotes the summation over all four-dimensional nonnegative integer coordinates (a, b, c, d) such that a + b + c + d = n, and x (0) = 1, x (n) = x(x + 1)(x + 2) · · · (x + n − 1) for all positive integers n.
According to this theorem, we can obtain the following corollaries: For any integer n > 0, we have

Corollary 2.
For any integer k > 0 and n > 0, we have

Corollary 4.
For any integer n > 0, we have

Several Simple Lemmas
To complete the proof of the theorem, we need the following two simple lemmas, which are essential to prove our main results.

Lemma 1.
For any integer r ∈ Z, we have where t 1 , t 2 , t 3 and t 4 are the four roots of the equation Proof. It is obvious that H n can be expressed the formula Since H 0 = H 1 = 0, H 2 = H 3 = 1, so we can get the system of equations On the other hand, we observe that Then note that (4) can also be written as Thus, we have Hence, by (2) and (5), we immediately obtain Now we have completed the proof of Lemma 1.

Lemma 2.
For a real number x ∈ R and any integer n ≥ 0, we have Proof. For any non-negative integers a, b, c and d, we have where {i, j, k, m} go through permutations of {a, b, c, d}.
Observe that the non-negative integers coordinates (a, b, c, d) with a + b + c + d = n is symmetrical, then we can obtain On the other hand, we have Then, applying (6) and (7), we obtain Lemma 2.

Proofs of the Main Results
In this section, we will prove our theorem and corollaries. For any real number x ∈ R, applying the properties of power series, we have x (n) n! t n , (|t| < 1), we note that t 1 , t 2 , t 3 and t 4 satisfy t 1 t 2 t 3 t 4 = −1, so