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American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
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The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
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Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
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The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

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On the Generalized Degenerate Tangent Numbers and Polynomials

Cheon Seoung Ryoo

Department of Mathematics, Hannam University, Daejeon, Korea

Abstract: In [7], Ryoo introduced the generalized tangent numbers and polynomials. In this paper, our goal is to give generating functions of the degenerate generalized tangent numbers and polynomials. We also obtain some explicit formulas for degenerate generalized tangent numbers and polynomials.

Keywords: generalized tangent numbers and polynomials, degenerate generalized tangent numbers and polynomials

Citation: Cheon Seoung Ryoo, On the Generalized Degenerate Tangent Numbers and Polynomials. Turkish Journal of Analysis and Number Theory, 3 (4). 104-107. January 2015. doi:10.12691/tjant-3-4-3

1. Introduction

In [2], L. Carlitz introduced the degenerate Bernoulli polynomials. Recently, Feng Qi et al. [3] studied the partially degenerate Bernoull polynomials of the first kind in p-adic field. In [9], Ryoo constructed the degenerate tangent numbers and polynomials. In this paper, we introduce the degenerate generalized tangent numbers and polynomials. We also obtain some interesting properties for degenerate generalized tangent numbers and polynomials. Throughout this paper, we always make use of the following notations: denotes the set of natural numbers and denotes the set of complex numbers. The tangent numbers are defined by means of the following generating function:

[picture] | (1.1)

Ryoo [6] defined tangent polynomials by multiplying on the right side of the Eq. (1.1) as follows:

[picture] | (1.2)

Let be Dirichlet's character with conductor with Then the generalized tangent numbers associated with associated with , are defined by the following generating function

[picture] | (1.3)

We now consider the generalized tangent polynomials associated with are also defined by

[picture] | (1.4)

When , above (1.3) and (1.2) will become the corresponding definitions of the tangent numbers and polynomials (see [6]). For more theoretical properties of the generalized tangent numbers and polynomials, the readers may refer to [7].

For a variable t, we consider the degenerate tangent polynomials which are given by the generating function to be

[picture] | (1.5)

When are called the degenerate tangent numbers(see [8]).

We recall that the classical Stirling numbers of the first kind and are defined by the relations (see [9,10])

respectively. Here denotes the falling factorial polynomial of order The numbers also admit a representation in terms of a generating function

[picture] | (1.6)

We also have

[picture] | (1.7)

The generalized falling factorial with increment is defined by

[picture] | (1.8)

for positive integer n, with the convention We also need the binomial theorem:

for a variable ,

[picture] | (1.9)

2. On the Degenerate Generalized Tangent Polynomials

In this section, we define the degenerate generalized tangent numbers and polynomials, and we obtain explicit formulas for them. Let be Dirichlet's character with conductor with Then the degenerate generalized tangent numbers associated with are defined by the following generating function

[picture] | (2.1)

When are called the generalized degenerate tangent numbers.

From (2.1) and (1.4), we note that

Thus, we get

From (2.1) and (1.5), we have

[picture] | (2.2)

Therefore, by (2.1) and (2.2), we obtain the following theorem.

Theorem 1. For , we have

For with we have

[picture] | (2.3)

By comparing coefficients of in the above equation, we have the following theorem.

Theorem 2. Let be Dirichlet's character with conductor with Then we have

(1)

(2)

From (2.1), we can derive the following relation:

[picture] | (2.4)

By comparing of the coefficients on the both sides of (2.4), we have the following theorem.

Theorem 3. For we have

By (2.1), we have

[picture] | (2.5)

By comparing of the coefficients on the both sides of (2.5), we have the following theorem.

Theorem 4. For we have

From (2.1), we have

[picture] | (2.6)

Therefore, by (2.6), we have the following theorem.

Theorem 5. For we have

From Theorem 5, we note that is a Sheffer sequence.

By replacing t by in (2.1), we obtain

[picture] | (2.7)

Thus, by (2.7) and (1.4), we have the following theorem.

Theorem 6. For we have

By replacing t by in (1.4), we have

[picture] | (2.8)

and

[picture] | (2.9)

Thus, by (2.8) and (2.9), we have the following theorem.

Theorem 7. For we have

References

[1]   | L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15(1979), 51-88.
[2]   | L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7(1956), 28-33.
[3]   | F. Qi, D. V. Dolgy, T. Kim, C. S. Ryoo, On the partially degenerate Bernoulli polynomials of the first kind, Global Journal of Pure and Applied Mathematics, 11(2015), 2407-2412.
[4]   | T. Kim, Barnes' type multiple degenerate Bernoulli and Euler polynomials, Appl. Math. Comput. 258(2015), 556-564.
[5]   | H. Ozden, I. N. Cangul, Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. 18(2009), no. 1, 41-48.
[6]   | C. S. Ryoo, A Note on the tangent numbers and polynomials, Adv. Studies Theor. Phys., 7(2013), no. 9, 447-454.
[7]   | C. S. Ryoo, Generalized tangent numbers and polynomials associated with p-adic integral on p, Applied Mathematical Sciences, 7(2013), no. 99, 4929-4934.
[8]   | C. S. Ryoo, Some identities on the (h; q)-tangent polynomials and Bernstein Polynomials, Applied Mathematical Sciences, 8(2014), no. 75, 3747-3753.
[9]   | C. S. Ryoo, Notes on degenerate tangent polynomials, to appear in Global Journal of Pure and Applied Mathematics, Volume 11, number 5(2015), pp. 3631-3637.
[10]   | P. T. Young, Degenerate Bernoulli polynomials, generalized factorial sums, and their applications, Journal of Number Theorey, 128(2008), 738-758.