1. Introduction
Recently, many mathematicians have studied in the area of the Bernoulli numbers, Euler numbers, Genocchi numbers, and tangent numbers (see [2,3,4,6,7,8]). Tangent numbers Tn and polynomials, Tn(x)(n ≥ 0), were introduced by Ryoo (see [5]). The tangent numbers Tn are defined by the generating function:
We introduce the tangent polynomials Tn(x) as follows:
In [6], Tangent numbers of higher order, are defined by means of the following generating function
The first few of them are
Nonlinear differential equations arising from the generating functions of special polynomials are studied by T. Kim and D. Kim in order to give explicit identities for special polynomials(see [1,4]). In this paper, we study differential equations arising from the generating functions of tangent numbers. We give explicit identities for the tangent numbers.
2. Differential equations associated with tangent numbers
In this section, we study linear differential equations arising from the generating functions of tangent numbers. Let
Then, by (2.1), we have
and
Continuing this process, we can guess that
Taking the derivative with respect to t in (2.4), we have
On the other hand, by replacing N by N + 1 in (2.4), we get
By (2.5) and (2.6), we have
Comparing the coefficients on both sides of (2.7), we obtain
and
In addition, by (2.4), we get
Thus, by (2.10), we obtain
It is not difficult to show that
Thus, by (2.12), we also get
From (2.8), we note that
and
For i = 2, 3, 4 in (2.9), we have
and
Continuing this process, we can deduce that, for 2 ≤ i ≤ N + 1,
Here, we note that the matrix ai(j)1≤i≤N+2,0≤j≤N+1 is given by
Now, we give explicit expressions for ai(N + 1). By (2.14) and (2.15), we get
and
Continuing this process, we have
Therefore, by (2.16), we obtain the following theorem.
Theorem 2.1. For N = 0, 1, 2, . . . , the functional equations
have a solution
where
Here is a plot of the surface for this solution. In Figure 1, we plot of the shape
FIGURE 1.The shape for the solution F(t)
for this solution.
From (1.1), we note that
From Theorem 1, (1.3), and (2.17), we can derive the following equation:
By comparing the coefficients on both sides of (2.18), we obtain the following theorem.
Theorem 2.2. For k,N = 0, 1, 2, . . . , we have
Let us take k = 0 in (2.19). Then, we have the following corollary.
Corollary 2.3. For N = 0, 1, 2, . . . , we have
참고문헌
- A. Bayad, T. Kim, Higher recurrences for Apostal-Bernoulli-Euler numbers, Russ. J. Math. Phys. 19 (2012), 1-10. https://doi.org/10.1134/S1061920812010013
- A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol 3. New York: Krieger, 1981.
- D.S. Kim, T. Kim, Some identities of Bell polynomials, Sci. China Math. 58 (2015), 2095-2104. https://doi.org/10.1007/s11425-015-5006-4
- T. Kim, D.S. Kim, Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl. 9 (2016), 2086-2098. https://doi.org/10.22436/jnsa.009.05.14
- C.S. Ryoo, A note on the tangent numbers and polynomials, Adv. Studies Theor. Phys. 7 (2013), 447 - 454. https://doi.org/10.12988/astp.2013.13042
- C.S. Ryoo, Multiple tangent zeta function and tangent polynomials of higher order, Adv. Studies Theor. Phys. 8 (2014), 457 - 462. https://doi.org/10.12988/astp.2014.4442
- C.S. Ryoo, A numerical investigation on the zeros of the tangent polynomials, J. App. Math. & Informatics 32(2014), 315-322. https://doi.org/10.14317/jami.2014.315
- S. Roman, The umbral calculus, Pure and Applied Mathematics, 111, Academic Press, Inc. [Harcourt Brace Jovanovich Publishes]. New York, 1984.
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