Binomial tribonacci sums

Abstract We derive expressions for several binomials sums involving a generalized tribonacci sequence. We also study double binomial sums involving this sequence. Several explicit examples involving tribonacci and tribonacci–Lucas numbers are stated to highlight the results.


Introduction
There is a dearth of tribonacci summation identities including binomial coefficients. Our goal in this paper is to derive several new binomial tribonacci sums such as In the above identities, n denotes a non-negative integer, s and p are arbitrary integers and G n is a generalized tribonacci number. The generalized tribonacci sequence G n = G n (c 0 , c 1 , c 2 ), n ≥ 0, is defined recursively by with initial values G 0 = c 0 , G 1 = c 1 , G 2 = c 2 not all being zero. Extension of the definition of G n to negative subscripts is provided by writing the recurrence relation as so that G n is defined for all integers n.
The most prominent representatives of G n and widely studied in the literature are G n (0, 1, 1) = T n the sequence of tribonacci numbers and G n (3, 1, 3) = K n the sequence of tribonacci-Lucas numbers (sequences A000073 and A001644 in [19], respectively).
The first few tribonacci numbers and tribonacci-Lucas numbers with positive and negative subscripts are given in Table 1.  Properties of (generalized) tribonacci sequences were investigated in the recent articles [1-4, 7, 8, 10, 12-18, 20, 21], among others. For instance, Janjić [16] found the remarkable combinatorial identity A generalized tribonacci number G n (c 0 , c 1 , c 2 ) is given by the Binet formula where α, β and γ are the distinct roots of the equation The coefficients A, B and C depend on the initial values and are determined by the system The Binet formulas for T n and K n are and , is a primitive cube root of unity. Tribonacci and tribonacci-Lucas numbers with negative indices can be accessed directly, using the following result.
In this article, we study binomial and double binomial sums with terms being a generalized tribonacci sequence. We derive closed forms for several such sums. We also prove a general binomial identity characterizing G an+b for a ≥ 1 and b an arbitrary integer.

Some auxiliary results
In this section we present some results that we will use in the sequel.

Lemma 2.2. We have
with identical relations for β and γ.

Identities from the binomial theorem and binomial transform
The next lemma will be the key ingredient to derive many results in this paper. For a proof and some applications to Horadam numbers, see [11].
Lemma 3.1. Let n and j be integers with 0 ≤ j ≤ n. Then, for each x, y ∈ C, we have We also mention the standard fact about sequences and their binomial transforms [5]: Let (a n ) n≥0 be a sequence of numbers and (b n ) n≥0 be its binomial transform. Then we have the following relations: Furthermore, if a 0 = 0 (so that b 0 = 0 too) the binomial pair exhibits the following properties: Proof. Use identity (9) in Lemma 3.1 with x = 1 and y = α 4 , taking note of Lemma 2.3.
From (19) and (20) we immediately obtain the following binomial tribonacci and tribonacci-Lucas relations.

Theorem 3.2. For non-negative integer n, any integer s, we have
where the values of δ, p, q and r as given in each column in Table 2. Proof. Each of the identities (5)-(8) can be written as (α p + δ) 3 = 2 q α r , where the values of δ, p, q and r in each case are as given in each column in Table 2. The identity of the theorem then follows from the binomial theorem and Lemma 2.3.

Lemma 3.2.
For non-negative integer n and real or complex z,
Setting s = 0 in Theorem 3.3, we immediately obtain the following.

Corollary 3.3. For non-negative integer n,
As special cases of formulas above we have:

Identities from the Waring formulas
Our next result provides two combinatorial identities for generalized tribonacci numbers involving binomial coefficients.
Lemma 4.1. The following identities hold for n ≥ 0 and real or complex x and y: and n/2 Formulas (25) and (26) are well-known in combinatorics and called Waring (sometimes Girard-Waring) formulas. The proof of these formulas can be found, for example, in [9].