In this paper, we first define the Gaussian modified Pell sequence, for n ≥ 2, by the relation = 2 −1 + −2 with initial conditions 0 = 1 ─ i and 1 = 1 + i. Then we give the definition of the Gaussian modified Pell polynomial sequence, for n ≥ 2, by the relation ( ) = 2 −1 ( ) + −2 ( ) with initial conditions 0 ( ) = 1─ xi and 1 ( ) = x + i. We give Binet's formulas, generating functions and summation formulas of these sequences. We also obtain some well-known identities such as Catalan's identities, Cassini's identities and d'Ocagne's identities involving the Gaussian modified Pell sequence and Gaussian modified Pell polynomial sequence. KeywordsModified Pell sequence, Modified Pell polynomial sequence, Gaussian modified Pell sequence, Gaussian modified Pell polynomial sequence.

In this paper, we investigate a generalization of modified Pell sequence, which is called (,)-modified Pell sequence. By considering this sequence, we define the matrix sequence whose elements are (,)-modified Pell numbers. Furthermore, we obtain Binet formulas, the generating functions and some sums formulas of these sequences. Finally, we give some relationships between (,)-Pell and (,)-modified Pell matrix sequences.

In this paper, we study the modified Pell polynomials. We first give the proof of the generating function of these polynomials. We then give the proof of the Binet formula for the modified Pell polynomials, which gives the th modified Pell polynomial. We also obtain some summation formula for these polynomials. In addition, we investigate some well-known identities including Catalan, Cassini, d'Ocagne and Gelin-Cesaro identities involving the modified Pell polynomials.

The authors of the paper [1] realized that since the concept of the modified Pell polynomials can be obtained as a special case of Horadam polynomials which introduced in [2], the paper [2] should be cited in introduction of [1]. For this reason, the sentence“Additionally, the modified Pell polynomials are defined recursively by the relation  = 2 +  where =1 and =x.”should be corrected as “Additionally, as a special case of Horadam polynomials introduced by Horzum and Kocer in [2], the modified Pell polynomials are defined recursively by the relation  = 2 +  where   = 1  and   = x.”

Bu çalışmanın amacı Gauss Bronz Lucas sayı dizisini tanıtmak ve incelemektir. İlk olarak Bronz Lucas sayılarını genişleterek Gauss Bronz Lucas sayılarını tanımladık. Daha sonra bu sayı dizisi için Binet formülü ve üreteç fonksiyonunu bulduk. Ayrıca Gauss Bronz Lucas sayıları ile ilgili bazı toplam formülleri ve matrisleri araştırdık. Son olarak, bu dizinin Binet formülünü dikkate alarak Catalan, Cassini ve d’Ocagne özdeşlikleri gibi bilinen eşitlikleri elde ettik.