Abstract

In this paper, we introduce a operator in order to derive a new generating functions of modified k- Pell numbers, Gaussian modified Pell numbers. By making use of the operator defined in this paper, we give some new generating functions for Bivariate Complex Fibonacci and Lucas Polynomials, modified Pell Polynomials and Gaussian modified Pell Polynomials.

2010 Mathematics Subject Classification. Primary 05E05; Secondary 11B39.

1. Introduction

The modified Pell numbers and Gaussian modified Pell numbers are the numbers of positive integers that have been studied over several years. These numbers are examples of a numbers defined by a recurrence relation of second order. It is well known that the modified Pell numbers is defined in ¹ by the following recurrence relation with initial conditions

In ² Tulay Yagmur and Nusret Karaaslan are defined the Gaussian modified Pell numbers by the recurrence relation for with initial conditions , and then they give the definition of the Gaussian modified Pell polynomials, for by the relation , with initial conditions and

Mustafa Asci and Esref Gurel are define and study the Bivariate Complex Fibonacci and Lucas Polynomials in ³. They give generating function, Binet formula, explicit formula and partial derivation of these polynomials. By defining these Bivariate Polynomials for special cases is the complex Fibonacci polynomials defined in ⁴ and is the complex Fibonacci numbers, and give the divisibility properties of Bivariate Complex Fibonacci Polynomials.

The Bivariate Complex Fibonacci Polynomials are defined by the following recurrence relation

with initial conditions and

The bivariate complex Lucas polynomials are defined by the following recurrence relation

with initial conditions and

In ⁵, the modified Pell polynomials are defined recursively by with initial conditions and

In this contribution, we are going to define an operator denoted by that formulates, extends and proves results based on our previous ones, see ^{6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}. In order to determine generating functions of modified k-Pell numbers, Gaussian modified Pell numbers, Bivariate Complex Fibonacci and Lucas Polynomials, modified Pell Polynomials and Gaussian modified Pell Polynomials, we use analytical means and series manipulation methods. In the sequel, we derive new symmetric functions and some new properties. We also give some more useful definitions which are used in the subsequent sections. From these definitions, we prove our main results given in Section 3.

2. Definitions and some Properties

In this section, we introduce a symmetric function and give some properties of this symmetric function. We also give some more useful definitions from the literature which are used in the subsequent sections.

We shall handle functions on different sets of indeterminates (called alphabets, though we shall mostly use commutative indeterminates for the moment). A symmetric function of an alphabet is a function of the letters which is invariant under permutation of the letters of . Taking an extra indeterminate , one has two fundamental series

The expansion of which gives the elementary symmetric functions and the complete symmetric functions

Let us now start at the following definition.

Definition 1: Let and be any two alphabets, then we give by the following form:

(2.1)

with the condition for (see ²⁶).

Corollary 1: Taking in (2.1) gives

(2.2)

Further, in the case or , we have

(2.3)

Thus,

(see ²⁶).

Definition 2: ²⁷ Let be any function on , then we consider the divided difference operator as the following form

Definition 3: ⁶ Given an alphabet the symmetrizing operator is defined by

3. Construction of Generating Functions of Some Numbers and Polynomials

The following proposition is one of the key tools of the proof of our main result. It has been proved in ⁷ for the completeness of the paper we state its proof here.

Proposition 1: Given an alphabet then

(3.1)

Based on the relationship we have

(3.2)

The substitutions in and we obtain

(3.3)

(3.4)

and we have the following results.

Corollary 2 : For , the new generating function of modified Pell numbers is given by

(3.5)

with .

Ÿ Put in the relationship we get the following corollary

Corollary 3: For , the new generating function of modified Pell numbers is given by

The substitutions in and we obtain

(3.6)

(3.7)

Multiplying the equation by and by , we obtain

Accordingly, we conclude the following Corollary.

Corollary 4: For , the new generating function of Gaussian modified Pell numbers is given by

with

The substitutions in and we obtain

(3.8)

(3.9)

and we have the following results.

Corollary 5: For , the new generating function of modified Pell Polynomial is given by

with

Multiplying the equation by and by we obtain

Accordingly, we conclude the following Corollary

Corollary 6: For , the new generating function of Gaussian modified Pell Polynomial is given by

with

Choosing and such that and substituting in and , we obtain

(3.10)

(3.11)

and we have the following Corollary.

Corollary 7: For the new generating function of Bivariate Complex Fibonacci is given by

with .

Multiplying the equation by and by we obtain

Accordingly, we conclude the following Corollary.

Corollary 8: For the new generating function of Bivariate Complex Lucas is given by

with

4. Conclusion

In this paper, by making use of Eq. (3.1), we have derived some new generating functions for the modified k- Pell numbers, Gaussian modified Pell numbers, Bivariate Complex Fibonacci and Lucas Polynomials, modified Pell Polynomials and Gaussian modified Pell Polynomials. The derived proposition and corollaries are based on symmetric functions and these numbers and polynomials.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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