Some Properties of Bi-Variate Bi-Periodic Lucas Polynomials

Abstract

The generalisation of Fibonacci sequence introduced by Edson in 2009. After the generalisation of Fibonacci sequence, Bilgici introduced generalized Lucas sequences. In 2016, Yilmaz and Coskun introduced generalisation of Fibonacci and Lucas polynomials which is known as bi-periodic Fibonacci polynomial and bi-periodic Lucas polynomials. In 2020, Verma and Bala defined bi-variate bi-periodic Fibonacci polynomials.  Now, We have defined Bi-variate Bi- periodic Lucas polynomials for  with initial conditions  by the recurrence relation  and  We have obtained generating function for defined polynomial and found n^th term of the . Investigated relationship between Bi-variate Bi-periodic Fibonacci and Bi-variate Bi- periodic Lucas polynomials. We derived some most popular identities like Cassini’s identity, Catalan’s identity, d’Ocagne’s identity and binomial sum. Convergence of two successive terms of Bi-variate Bi-periodic Lucas polynomial  is also discussed.