On some identities for the DGC Leonardo sequence

: In this study, we examine the Leonardo sequence with dual-generalized complex ( DGC ) coefficients for p ∈ R . Firstly, we express some summation formulas related to the DGC Fibonacci, DGC Lucas

to Fibonacci and Lucas numbers in [4,11,18,24,42].The Fibonacci sequence is defined by the following linear recurrence relation with the initial values F 0 = 0 and F 1 = 1.The Fibonacci sequence has been generalized in many ways, some by preserving the initial values, and others by preserving the recurrence relation.The Lucas sequence is defined recursively by: with the initial values L 0 = 2 and L 1 = 1.As it seen from equations ( 1) and ( 2), the Fibonacci and Lucas sequences satisfy the same recurrence relation with different initial values.The Binet's formulas of the Fibonacci and Lucas sequences are as follows, respectively: and where α = 1+ are roots of the characteristic equation λ 2 − λ − 1 = 0, [24].These two sequences have more in common than their recursive structure.
The Leonardo sequence is another integer sequence which is related to the Fibonacci sequence and also to the Lucas sequence.The Leonardo sequence is given by the following non-homogeneous recurrence relation: with the initial values Le 0 = Le 1 = 1.The homogeneous recurrence relation of the Leonardo sequence is Le n+1 = 2Le n − Le n−2 , n ≥ 2 (6) with the initial values Le 0 = Le 1 = 1 and Le 2 = 3, [5].The Binet formula of the Leonardo sequence is There are many well-known and established relations between the Fibonacci, Lucas, and Leonardo sequences.For a positive integer n, the fundamental relations between them are as follows [5]: and the summation formulas are (see [5,7]): Characteristic properties and some generalizations of the Leonardo sequence have been studied by researchers.In 2019, the Leonardo sequence was examined in detail by P. Catarino and A. Borges in [5]. A. G. Shannon studied the generalization of the Leonardo sequence, [33].P. Catarino and A. Borges defined incomplete Leonardo numbers and analyzed their recurrence relations, some properties, and the generating functions in [6].The generating matrices and the matrix form of the Leonardo sequence were investigated in [43].Y. Alp and E. G. Köc ¸er introduced the hybrid Leonardo sequence in [2].The relations among Fibonacci, Lucas, and Leonardo numbers were given in [3].In 2021, the relations between the hybrid Leonardo sequence and the hybrid Fibonacci sequence, Catalan's, Cassini's, and d'Ocagne's identities were presented in [26].The elliptical biquaternion Leonardo sequence was discussed in [27], and the hyperbolic Leonardo sequence was examined in [44].In 2022, the hybrid quaternion Leonardo sequence and their identities were presented in [28].The real and complex generalizations of the Leonardo sequence were introduced in [34]. A. Karatas ¸discussed the complex Leonardo sequence and their special identities in [22].S. Ö. Karakus ¸, S. K. Nurkan and M. Tosun characterised the hyper-dual Leonardo sequence in [21].M. Shattuck provided combinatorial proofs of some important identities satisfied by the generalized Leonardo numbers in [35].G.Y. S ¸entürk presented a brief study on the Leonardo sequence with dual-generalized complex coefficients in [40].In 2023, S. Kaya Nurkan and I. A. Güven combined the Leonardo sequence and dual quaternions, [30].Y. Soykan defined the modified p-Leonardo, p-Leonardo-Lucas, and p-Leonardo sequences as special cases of the generalized Leonardo sequence in [38].H. Özimamoglu introduced the hybrid q-Leonardo sequence by using q-integers in [31].E. Tan and H. H. Leung investigated the Leonardo p-sequence, and incomplete Leonardo p-sequence in [41].Z. İs ¸bilir, M. Akyigit and M. Tosun investigated the Pauli-Leonardo quaternion sequence in [19].A. Karakas ¸defined the concept of dual Leonardo numbers in [23].
The set C p is a vector space over R. All these special number systems have led to the construction of some 4-dimensional number systems.By utilizing the generalized complex and dual numbers, the dual-generalized complex (DGC) numbers are introduced in [15].The set of DGC numbers was introduced as: and discussed in [15].Any DGC number is of the form It gives the set of: • dual-complex numbers with elements ã = a 1 +a 2 i+a 3 ε+a 4 iε for p = −1 (see [8,9,25,29]), • hyper-dual numbers with elements ã = a 1 + a 2 ϵ + a 3 ε + a 4 ϵε for p = 0 (see [10,12,13]), • dual-hyperbolic numbers with elements ã = a 1 + a 2 j + a 3 ε + a 4 jε for p = 1 (see [1,25]).
The set of DGC numbers forms an associative and commutative ring with unity and a vector space of dimension 4 over R, [15].
In this study, inspired by the theory of the hypercomplex sequences, we examined the concept of the DGC Leonardo sequence for p ∈ R. We gave the characteristic formulas, involving d'Ocagne's, Catalan's, Cassini's, and Tagiuri's identities for the DGC Leonardo sequence.

Preliminaries
In this section, we follow [40] in presenting the basic notions of the DGC Leonardo sequence.
Table 1.Nomenclature of the sequences The sequence The general term (n-th term) The Fibonacci sequence F n The Lucas sequence L n The Leonardo sequence Le n The DGC Fibonacci sequence Fn The DGC Lucas sequence Ln The DGC Leonardo sequence Le n In the following Table 2, the several terms of the Fibonacci (see A000045 in [36]), Lucas (see A000032 in [36]), Leonardo (see A001595 in [36]), DGC Fibonacci, DGC Lucas, and DGC Leonardo sequences are given, respectively.
3 Some new identities for the DGC Leonardo Sequence In the following theorems, we touch only a new aspect to the DGC Leonardo sequence.
Theorem 3.1.For a positive integer n, the summation formulas related to the DGC Fibonacci, DGC Lucas, and DGC Leonardo sequences are as follows: 1. . . . .
Proof. 3. Using Equation ( 14), we have the following equations: We thus get: We thus get: A similar proof can be used to verify the other items.

Main results
In this subsection, we formulate the order-2 formulas for the DGC Leonardo sequence.
Theorem 3.3.For n and m positive integers, with m ⩾ n, the following identities are true: 1. Le Proof.The main idea of the proof is to take the relation Le n = 2 Fn+1 − 1 (see Theorem 2.1).

By using the relations
and (see Theorem 2, items 2 and 3, in [16]), we see that We conclude from F n L n = F 2n (see [24]) that Le 3. From the DGC Fibonacci recurrence relation, we have Theorem 3.4.For positive integers n and r, with n ≥ r, the general Catalan's identity for the DGC Leonardo sequence is as follows: Proof.We give two different proofs of the theorem.
Proof 1: By using the Binet's formulas in equations ( 3) and ( 16) for the Fibonacci and DGC Leonardo sequences, respectively, we deduce that: We conclude from α and β that αβ = −1, hence that and finally that Proof 2: From Theorem 2.1, it follows that (see Theorem 2, item 4 in [16] with m → n + 1 and n → n + 1), implies that In what follows, α β = (1 − p) + J + 3(1 − p)ε + 3Jε.Theorem 3.5.For a positive integer n, the general Cassini's identity (sometimes called Simson's identity) for the DGC Leonardo sequence is as below: Proof.Taking r → 1 in the previous theorem, we obtain the Cassini's identity.Proof.We give two different proofs of the theorem.
By considering αβ = −1 and the Binet's formula in (3) for the Fibonacci sequence, the proof is straightforward.
Proof 2: Applying Le n = 2 Fn+1 − 1 (see Theorem 2.1) to the left-hand side gives From Equation (17) (see Theorem 2, item 4 in [16] with m → m + 1, n → n + 2 and r → 1), and the recurrence relation of the DGC Fibonacci sequence, we obtain that: This completes the proof.Theorem 3.10.For positive integers n and m, with n ≥ m, the following identities hold: 1 Proof.
1. Writing the Binet's formulas for the DGC Fibonacci sequence in Equation ( 11) and the DGC Leonardo sequence in Equation ( 16) into the left-hand side, we have The proof is completed by using αβ = −1 and the Binet's formula for the Fibonacci sequence (3).
2. Similarly, we first apply the Binet's formulas for the DGC Fibonacci sequence in (11) and the DGC Leonardo sequence in (16) to the left-hand side.We thus get

Conclusion
In this paper, the order-2 relations for the DGC Leonardo sequence are computed for p ∈ R. The advantage of implementing this construction is that the dual-complex, the hyper-dual, and the dual-hyperbolic Leonardo sequences can be carried out for p ∈ {−1, 0, 1}, and J ∈ {i, ϵ, j}, respectively (see Table 3).Table 3. Special cases of the DGC Leonardo sequence The sequence type The Leonardo sequence J p Condition

Theorem 3 . 7 .
For positive integers n, m, r and s, with r ⩾ s, the special case of the Tagiuri identity for the DGC Leonardo sequence is as below: Le n+r Le n+s − Le n Le n+r+s = 4 5 α β(−1) n+1 (L r+s − (−1) s L r−s ) + 1( Le n + Le n+r+s − Le n+r − Le n+s ).

Theorem 3 . 11 .
The proof is completed by considering DGC Lucas sequence statement in Equation(12).For positive integers k, m and s, with m ≥ k and m ≥ s, the following identity holds:Le m+k Le m−k − Le m+s Le m−s = 4α β((−1) m−k F 2 k − (−1) m−s F 2 s ) + 1( Le m+s + Le m−s − Le m+k − Le m−k ).

Table 2 .
Several terms for the Fibonacci, Lucas, Leonardo, DGC Fibonacci, DGC Lucas, and DGC Leonardo sequences Theorem 2.1.For a positive integer n, the fundamental relations between the DGC Fibonacci, Lucas, and Leonardo numbers are: