Abstract

The main of this paper is to define and investigate a new class of the degenerate poly-Frobenius-Genocchi polynomials with the help of the polyexponential functions. In this paper, we define the degenerate poly-Frobenius-Genocchi polynomials of complex variables arising from the modified polyexponential functions, and establish some explicit expressions for these polynomials. Meanwhile, some interesting connections between these polynomials and some other special polynomials are also showed.

1. Introduction

Throughout this paper, N denotes the set of natural numbers, denotes the set of nonnegative integeres, R denotes the set of real numbers and C denotes the set of complex numbers. We begin by introducing the following definitions and notaitions ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}.

The Frobenius-Euler polynomials are defined by ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18};

(1.1)

where and

When x=0, are called the Frobenius-Euler numbers.

The Genocchi polynomials are defined by ^{11, 12, 14})

(1.2)

When x=0, are called the Genocchi numbers.

The Frobenius-Genocchi polynomials are defined by ¹⁸

(1.3)

For u=-1, and x=0, are called the Frobenius-Genocchi numbers.

The degenerate exponential function is defined by ^{3, 4, 5, 6, 7, 8, 9, 10, 11} with

and

(1.4)

where and

For and k nonnegative integer, the degenerate λ-Stirling polynomials of the second kind are defined by ⁵

(1.5)

Note that

From (1.4), we get

(1.6)

where

Using (1.4) and (1.6), we note that

The degenerate Stirling numbers of the first kind are defined by ^{3, 4, 5, 6, 7, 8, 9, 10}

(1.7)

Note here that where are the Stirling numbers of the first kind given by ⁵

(1.8)

The degenerate Stirling numbers of the second kind are defined by ^{3, 4, 5, 6, 7, 8, 9, 10}

(1.9)

Observe that where are the Stirling numbers of the second kind given by ⁵

(1.10)

The degenerate Bernoulli polynomials of the second kind are given by ^{6, 8}

(1.11)

Note that where are the Bernoulli polynomials of the second kind given by ⁶

(1.12)

2. Degenerate Poly-Frobenius-Genocchi Numbers and Polynomials

In this section, we introduce and investigate the modified polyexponential functions. We give some identities and explicit relations for the modified degenerate polyexponential functions. We define the degenerate poly-Frobenius-Genocchi polynomials. Also, we give some relations and identities for these polynomials.

In ², Boyadzhiev introduced the polyexponential function, Kim et al. in ^{6, 7} considered and investigated the polyexponential functions and the degenerate polyexponential functions.

The polyexponential functions are defined by ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 14}

(2.1)

For k=1,

The modified degenerate polyexponential functions are given by ^{3, 4, 5, 6, 7, 8, 9, 10, 11, 14}

(2.2)

Note that

For and by means of the modified degenerate polyexponential functions. We define the degenerate poly-Frobenius-Genocchi polynomials by the following generating functions.

(2.3)

When x=0, are called the degenerate poly-Frobenius-Genocchi numbers, where is the compositional inverse of satisfying

For k=1 and u=-1, we get the degenerate Genocchi polynomials

From (2.3), we can write the following equations

(i)

(ii)

(iii)

By (1.8) and (2.2), we get

(2.4)

Using (2.3) and (2.4), we get

By using Cauchy product and comparing the coefficients of the above equations, we have the following theorem.

Theorem 1. For we have

From (2.3), we write as

(2.5)

Comparing the coefficients of both sides in (2.5).

We have the following theorem.

Theorem 2. For we have

From (2.2), we note that

(2.6)

Thus, by (2.5), we get

where with

From (1.11), (2.3) and (2.6), for k=2

From the last equations, we have the following theorem.

Theorem 3. For we have

where is degenerate Frobenius-Euler numbers.

Recently, Masjed-Jamai et al. in ¹³ and Srivastava et al. in ^{15, 16} introduced a new type parametric Euler numbers and polynomials as

and

where

and

3. Degenerate Poly-Frobenius-Genocchi Polynomials of Complex Variables

In this section, we define the Frobenius-Genocchi polynomials of the complex variables. We consider the degenerate cosine function and the degenerate sine function. Using the degenerate cosine function and the degenerate sine function, we introduce the cosine degenerate poly-Frobenius-Genocchi polynomials and the sine degenerate poly-Frobenius-Genocchi polynomials.

From (2.3), we write as

(3.1)

and

(3.2)

By (3.1) and (3.2), we get

(3.3)

and

(3.4)

Using (1.4), we define the degenerate cosine-functions and the degenerate sine-functions as

(3.5)

and

(3.6)

where and

Now, we define the cosine degenerate poly-Frobenius-Genocchi polynomials and the sine degenerate poly-Frobenius-Genocchi polynomials, respectively;

(3.7)

and

(3.8)

From (1.4), we write

Using (3.5) and (3.6), we get

(3.9)

and

(3.10)

By (1.4), (3.9) and (1.4), (3.10), we have the following equations, respectively,

(3.11)

and

(3.12)

From (3.7) and (3.11), we write

Using the Cauchy product and comparing the coefficients, we have

(3.13)

From (3.8) and (3.11), similarly, we have

(3.14)

From (3.13) and (3.14), we have the following theorems.

Theorem 4. The following relations hold true:

and

Now, we define the degenerate two parametric and polynomials, respectively,

(3.15)

and

(3.16)

From (1.4) and (3.9), we get

Similarly, (1.4) and (3.10), we get

From (2.4), (3.7) and (3.11), we write

The left hand side of this equation is

(3.17)

The right hand side of this equation is

(3.18)

From (3.17) and (3.18), we get

(3.19)

Similarly, (2.4), (3.8) and (3.12)

(3.20)

Theorem 5. The following relations hold true:

and

From (1.6) and (3.7),

(3.21)

From (1.6) and (3.8), we get

(3.22)

Comparing the coefficients of both sides the equations (3.21) and (3.22), we have the following theorem.

Theorem 6. The following relations hold true:

and

Now, for our use in the present investigation, we find the expressions of and

From (3.5), we get

(3.23)

Putting (3.23), we get

By (3.6), we get

(3.24)

Setting (3.24), we get

From (3.15) and (3.23), we write

(3.25)

Using (3.16) and (3.24), we write

(3.26)

By using Cauchy product above the equations (3.25) and (3.26), we have the following theorem.

Theorem 7. The following relations hold true:

(3.27)

and

(3.28)

Setting and in (3.27) and (3.28), we have respectively,

and

From (3.7) and (3.22), we write

(3.29)

Using (3.8) and (3.24), we write

(3.30)

Using Cauchy product (3.29) and (3.30), we have the following theorem.

Theorem 8. The following relations hold true:

(3.31)

and

(3.32)

Putting and in (3.31) and (3.32), respectively, we have

and

4. Conclusion

Kim and Kim ⁷ considered the polyexponential and unipoly functions. Kim et al. ^{3, 11} defined and investigated the new type modified degenerate polyexponential function, the degenerate poly-Bernoulli polynomials and the degenerate poly-Genocchi polynomials. Motivated by these studying, we introduce the degenerate poly-Frobenius-Genocchi polynomials of the complex variables. We also give their some interesting properties and identities. As one of our future projects, we would like to continue to do researcher on degenerate versions of various special numbers and polynomials.

Acknowledgements

The present investigation was supported, in part, by the Scientific Research Project Administration of the University of Akdeniz.

References