Some new identities of a type of generalized numbers involving four parameters

Abstract

This article deals with a Horadam type of generalized numbers involving four parameters. These numbers generalize several celebrated numbers in the literature such as the generalized Fibonacci, generalized Lucas, Fibonacci, Lucas, Pell, Pell-Lucas, Fermat, Fermat-Lucas, Jacobsthal, Jacobsthal-Lucas, balancing, and co-balancing numbers. We present some new identities concerned with the generalized numbers of four parameters. An explicit expression for these numbers is developed, and a mixed recurrence relation between two certain families of the generalized numbers is given, and after that, some new identities are presented and proved. A large number of identities between several celebrated numbers are obtained as special cases of our developed ones. Furthermore, some of the identities that were previously published in other articles can be deduced as special ones of our new identities.

Keywords

• generalized fibonacci and generalized lucas numbers
• lucas and fibonacci numbers
• recurrence relation
• zeilberger's algorithm