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On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter

Burak KURT

Akdeniz University, Educations Faculty, Department of Mathematics, Antalya-TURKEY

Abstract: Many kinds of generalizations of these polynomials and numbers have been presented in the literature (see [1,2]). Bayad and Hamahata in [1] introduced and investigated the poly-Bernoulli polynomials and proved some relations for these polynomials. Cenkci and Kamatsu in [3] are defined q-parameter poly-Bernoulli numbers. They proved some relations between these polynomials. Kim et al. in ([4,5]) defined poly-Genocchi polynomials and gave the some properties for the multiple-poly-Bernoulli numbers and multiple-zeta values. We introduce the Hermite-based poly-Genocchi polynomials with a q-parameter. After, we give and investigate some properties and identities for these polynomials. Furthermore, we prove closed formula and two explicit relations.

Keywords: the Genocchi polynomials and numbers, the 2-variable Hermite-Kampé de Feriét polynomials, the polylogarithm function, the poly-Genocchi polynomials

Received September 28, 2019; Revised October 26, 2019; Accepted 28, 2019

Citation: Burak KURT, On the Hermite-Based Poly-Genocchi Polynomials with a q-Parameter. Turkish Journal of Analysis and Number Theory, 7 (6). 145-149. January 2019. doi:10.12691/tjant-7-6-2

1. Introduction

As usual, throughout this paper, N denotes the set of natural numbers, denotes the set of nonnegative integers, Z denotes the set of integer numbers, R denotes the set of real numbers and C denotes the complex numbers.

In the usual notations, let and denote respectively, the classical Bernoulli polynomials, the classical Euler polynomials and the classical Genocchi polynomials defined by the following generating functions;

[picture] | (1)

[picture] | (2)

and

[picture] | (3)

Also, let x=0,

Where and are respectively, the classical Bernoulli numbers, the classical Euler numbers and the classical Genoccchi numbers.

The 2-variable Hermite-Kampé de Feriét polynomials are defined by (see [6,7,8])

[picture] | (4)

Let the k-th polylogarithm function is defined by (see [1,6,9])

[picture] | (5)

when k=1, In the case are the rational functions:

Further information about poly-logarithm function and polynomials (see [1,4]).

Cenkci et al. in [3] defined the weighted Stirling numbers of the second kind as

[picture] | (6)

Duran et al. in [6] defined the Hermite-based 𝜆-Stirling polynomials of the second kind as

[picture] | (7)

The special values of the (6) are given in [6]. Let We define the Hermite-based poly-Genocchi polynomials with a q-parameter by the following generating functions:

[picture] | (8)

For x=y=0, we get which is called a new class of the Hermite-based poly-Genocchi number with a q-parameter. Some special cases of are following remarks.

Remark. For y=0, we have called the Hermite-based poly-Genocchi polynomials with a q-parameter.

Remark. For q=1, reduces to the Hermite-based poly-Genocchi polynomials.

Remark. For q=1 and y=0, reduces to the poly-Genocchi polynomials.

Remark. When q=k=1 and y=0, we obtain the classical Genocchi polynomials.

Srivastava and Srivastava et al. in ([10,11]) investigated some properties and proved some theorems for the Bernoulli, Euler and Genocchi polynomials. D. S. Kim et al. in ([12,13]) introduced the poly-Bernoulli polynomials and gave some recurrences relations and idebtities. Cenkci et al. in [3] gave the poly-Bernoulli polynomials with a q-parameter. Kurt [7] gave the poly-Genocchi polynomials with a q-parameter. Duran et al. in ([6,14]) considered the (p,q)-Hermite polynomials and the (p,q)-Euler polynomials.

2. Main Theorems

In this section, we give some basic identites and relations for the Hermite-based poly-Genocchi polynomial with a q-parameter. Further we give closed formula and explicit relation for these polynomials.

Theorem. The Hermite-based poly-Genocchi polynomials with a q-parameter satisfy following relation:

and

The proof of this theorem is easily obtain from the Hermite-based poly-Genocchi polynomials definition.

Theorem. The following relation holds true:

[picture] | (9)

Proof. By (4) and (8), we write as

L. H. S of this equation is

[picture] | (10)

R. H. S of this equation is

[picture] | (11)

From (10) and (11), we obtain (9).

Theorem. There is the following relation between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Euler polynomials:

[picture] | (12)

Proof. By (2) and (8), we write

By using Cauchy product and comparing the coefficients of we have (12).

Theorem. There is the following relation between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Bernoulli polynomials:

[picture] | (13)

Proof. From (1) and (8), we write as

By using Cauchy product and comparing the coefficients of we have (13).

Theorem. The following relation holds true:

[picture] | (14)

Proof. By (8),

From here, we have (14).

Theorem. There is following relationship between the Hermite-based poly-Genocchi polynomials with a q-parameter and the Stirling numbers of the second kind as:

[picture] | (15)

Proof. By (16) and (8), we write as:

L. H. S. of this equation is

[picture] | (16)

R. H. S. of this equation

By using Cauchy product, we have

[picture] | (17)

From (16) and (17), we get (15).

Theorem. The following relation holds true:

[picture] | (18)

Proof. By (8),

[picture] | (19)

We replace t by t+u in (19)

From this equation, we write as

[picture] | (20)

In the last equation, we replace x by v, we get

[picture] | (21)

By (12) and (13), we write

[picture] | (22)

Now, by applying the following known series identity [[19], p. 52, Eq. 1. 6 (2)]

[picture] | (23)

in the right hand side of (23), we get

[picture] | (24)

Finally, upon first replacing n by n-p and m by m-r by using the Cauchy product in the left hand side of the above equation (24) and comparing the coefficients of and On both sides of the resulting equation, we have (18).

Theorem (Closed Formula) The following relation holds true:

[picture] | (25)

Proof. By replacing k by (-k) in (8), we get

[picture] | (26)

For 𝜆=1 and j=2 in (8), we get

[picture] | (27)

We put the equation (8) and (27) in (26). We have

From the last equation, comparing the coefficients of and we have (25).

Acknowledgements

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

References

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