On a new generalization of Fibonacci and Lucas $p$-triangles

Abstract

In this study, we introduce a new generalization of Fibonacci and Lucas $p$-triangles, which also provides a novel extension of the well-known Pascal's and Lucas triangles. The primary motivation for this investigation is to derive explicit formulas for the bi-periodic Fibonacci and Lucas $p$-numbers. To achieve this, a generalization of binomial coefficients is derived and several of their properties, including recurrence relations, the generating function, and convolution identity are presented. Additionally, as an application of these triangles, we define bi-periodic incomplete Fibonacci and Lucas $p$-numbers and state several of their properties.