Abstract
In this paper, we introduce a operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers and Gaussian Jacobsthal polynomials. In the paper Al4, Al5, a second-order linear recurrence sequence (U_{n}(a,b;p,q))_{n≥0} or briefly (U_{n})_{n≥0} is considered by the recurrence relation: U_{n+2}=pU_{n+1}+qU_{n}, with the initial conditions U₀=a and U₁=b, where a,b∈ℂ and p,q∈ℤ₊ for n≥0.
Highlights
In the paper [9, 10], a second-order linear recurrence sequence (Un(a; b; p; q))n 0 or brie‡y (Un)n 0 is considered by the recurrence relation: Un+2 = pUn+1 + qUn; with the initial conditions U0 = a and U1 = b, where a; b 2 C and p; q 2 Z+ for n 0
We introduce a operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers
The derived theorems are based on symmetric functions and products of these numbers and polynomials
Summary
Symmetric functions, generating functions, Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Pell polynomials. The Gaussian Jacobsthal and Gaussian Jacobsthal Lucas polynomials GJn(x) and Gjn(x) are de...ned and studied by authors [15]. They give generating function, Binet formula, explicit formula, Q matrix, determinantal representations and partial derivation of these polynomials. Oz are de...ned in 2016 the Gaussian Pell and Gaussian Pell-Lucas numbers They give generating functions and Binet formulas of Gaussian Pell and Gaussian Pell-Lucas numbers. The authors in [16] de...ned Gaussian Pell polynomials, they give the generating functions and.
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