Abstract
In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found.
Highlights
In literature, the Fibonacci and Lucas numbers have been studied extensively and some authors tried to enhance and derive some directions to mathematical calculations using these special numbers [1,2,3]
Ozdemir and Simsek [14] give the family of two-variable polynomials, reducing some well-known polynomials and obtaining some properties of these polynomials
In light of these polynomials, we introduce the families of three-variable polynomials with the new generalized polynomials reduced to the generating functions of the famous polynomials and numbers in literature
Summary
The Fibonacci and Lucas numbers have been studied extensively and some authors tried to enhance and derive some directions to mathematical calculations using these special numbers [1,2,3]. The tribonacci numbers Tn for any integer n > 2 are defined via the following recurrence relation. By way of the initial values K0 = 3, K1 = 1, and K2 = 3, the tribonacci-Lucas numbers Kn are given by the recurrence relation. By dint of the above extensions, the tribonacci and tribonacci-Lucas numbers are introduced with the help of the following generating functions, respectively:. Ozdemir and Simsek [14] give the family of two-variable polynomials, reducing some well-known polynomials and obtaining some properties of these polynomials In light of these polynomials, we introduce the families of three-variable polynomials with the new generalized polynomials reduced to the generating functions of the famous polynomials and numbers in literature. We give the interesting applications of these polynomials
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