Some identities of bivariate Pell and bivariate Pell-Lucas polynomials

Year 2023, Volume: 4 Issue: 2 , 90 - 99 , 31.12.2023

Yashwant Panwar

https://doi.org/10.54559/jauist.1395447

https://izlik.org/JA73YF28LL

Abstract

In this paper, we obtain some identities for the bivariate Pell polynomials and bivariate Pell-Lucas polynomials. We establish some sums and connection formulas involving them. Moreover, we present its two cross two matrices representation and find some of its properties, such as the b^th power of the matrix. We finally derive the identities by using Binet’s formula, generating function, and induction method.

Keywords

Bivariate Pell polynomials , bivariate Pell-Lucas polynomials , Binet’s formula , generating function

References

• M. Catalani, Some formulae for bivariate Fibonacci and Lucas polynomials (2004) 9 pages, https://arxiv.org/abs/math/0406323.
• M. Catalani, Identities for Fibonacci and Lucas polynomials derived from a book of Gould (2004) 7 pages, https://arxiv.org/abs/math/0407105.
• M. Catalani, Generalized bivariate Fibonacci polynomials (2004) 11 pages, https://arxiv.org/abs/math/0211366.
• H. Belbachir, F. Bencherif, On some properties of bivariate Fibonacci and Lucas polynomials, Journal of Integer Sequences 11 (2) (2008) 08.2.6 10 pages.
• N. Tuğlu, E. G. Koçer, A. Stakhov, Bivariate Fibonacci like p-polynomials, Applied Mathematics and Computation 217 (24) (2011) 10239–10246.
• S. Halıcı, Z. Akyüz, On sum formulae for bivariate Pell polynomials, Far East Journal of Applied Mathematics 41 (2) (2010) 101–110.
• N. Saba, A. Boussayoud, Complete homogeneous symmetric functions of Gauss Fibonacci polynomials and bivariate Pell polynomials, Open Journal of Mathematical Sciences 4 (1) (2020) 179–185.
• T. Machenry, A subgroup of the group of units in the ring of arithmetic functions, Rocky Mountain Journal of Mathematics 29 (3) (1999) 1055–1065.
• T. Machenry, Generalized Fibonacci and Lucas polynomials and multiplicative arithmetic functions, Fibonacci Quarterly 38 (2) (2000) 167–173.
• T. Machenry, G. Tudose, Reflections on symmetric polynomials and arithmetic functions, Rocky Mountain Journal of Mathematics 35 (3) (2006) 901–928.
• T. Machenry, K. Wong, Degree k linear recursions mod(p) and number fields, Rocky Mountain Journal of Mathematics 41 (4) (2011) 1303–1327.