Some results on convolved (p, q)-Fibonacci polynomials

Abstract

In this paper, based on the (p, q)-Fibonacci polynomials $u_n(x)$ and (p, q)-Lucas polynomials $v_n(x)$, we introduce the convolved (p, q)-Fibonacci polynomials $u_n^{(r)}(x)$, which generalize the convolved Fibonacci numbers, the convolved Pell polynomials, and the Gegenbauer polynomials. We give the expressions, expansions, recurrence relations and differential recurrence relations of $u_n^{(r)}(x)$, and establish the relations between $u_n^{(r)}(x)$, $u_n(x)$ and $v_n(x)$. Moreover, we also study the determinantal representations of $u_n^{(r)}(x)$ and $v_n(x)$, and present an algebraic interpretation of the polynomials $u_n^{(r)}(x)$.

Keywords:

• (p, q)-Fibonacci polynomials
• (p, q)-Lucas polynomials
• convolved (p, q)-Fibonacci polynomials
• combinatorial identities
• determinants

AMS Classifications::

• 05A15
• 05A19
• 11B39

Acknowledgments

The authors would like to thank the anonymous referee for his (her) valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by the Zhejiang Provincial Natural Science Foundation of China [grant LY13A010016], and the ‘521’ Talents Program of Zhejiang Sci-Tech University [grant 11430132521303].