 Research
 Open Access
Computation of Nevanlinna characteristic functions derived from generating functions of some special numbers
 Serkan Araci^{1}Email authorView ORCID ID profile and
 Mehmet Acikgoz^{2}
https://doi.org/10.1186/s136600181722y
© The Author(s) 2018
 Received: 22 December 2017
 Accepted: 5 June 2018
 Published: 7 June 2018
Abstract
In the present paper, firstly we find a number of poles of generating functions of Bernoulli numbers and associated Euler numbers, denoted by \(n ( a,\mathbf{B} ) \) and \(n ( a,E ) \), respectively. Secondly, we derive the mean value of a positive logarithm of generating functions of Bernoulli numbers and associated Euler numbers shown as \(m ( 2\pi,\mathbf{B} ) \) and \(m ( \pi,E ) \), respectively. From these properties, we find Nevanlinna characteristic functions which we stated in the paper. Finally, as an application, we show that the generating function of Bernoulli numbers is a normal function.
Keywords
 Meromorphic function
 Poisson–Jensen formula
 Nevanlinna characteristic function
 Normal function
 Bernoulli numbers
 Associated Euler numbers
 Generating function
MSC
 11B68
 30D30
 30D35
1 Introduction and preliminaries
In the mathematical field of complex analysis, Nevanlinna theory deals with the theory of meromorphic functions. It was constructed in 1925 by Finnish mathematician Rolf Herman Nevanlinna (October 22, 1895–May 28, 1980), who made significant contributions to complex analysis. Because of devising of R. Nevanlinna, Hermann Weyl has called it “one of the few great mathematical events of (the twentieth) century” [1]. In fact, Nevanlinna theory plays an important role in transcendental meromorphic functions, analytic theory of differential and functional equations, holomorphic dynamics, minimal surfaces, and complex hyperbolic geometry, which deals with generalizations of Picard’s theorem to higher dimensions, cf. [1–8] and the references cited therein.
We now begin with the properties of Nevanlinna theory.
Theorem 1
Definition 1
Proposition 1
Proposition 2
Definition 2
Notice that the positive logarithm defined above is a continuous function of nonnegative on \(( 0,\infty ) \).
Corollary 1
Theorem 2
Definition 3
Theorem 3
The works on special numbers and polynomials have a very long history. In fact, special numbers and polynomials play a significantly important role in the progress of several fields of mathematics, physics, and engineering. They have many algebraic operations. That is, because of their finite evaluation schemes and closure under addition, multiplication, differentiation, integration, and composition, they are richly utilized in computational models of scientific and engineering problems. For more information related to special numbers and polynomials, see [9–11] and the references cited therein.
By this motivation, we find a number of poles of generating functions of Bernoulli numbers and associated Euler numbers, denoted by \(n ( a,\mathbf {B} ) \) and \(n ( a,E ) \), respectively. After that, we derive the mean value of a positive logarithm of generating functions of Bernoulli numbers and associated Euler numbers shown as \(m ( 2\pi,\mathbf{B} ) \) and \(m ( \pi ,E ) \), respectively. From these properties, we find Nevanlinna characteristic functions which we stated in the following parts. In the final part of this paper, as an application, we show that the generating function of Bernoulli numbers is a normal function.
2 Nevanlinna characteristic function of generating function of Bernoulli numbers
It means \(B ( z ) \) has a removable singular point at \(z=0\). Then we have the following corollary.
Corollary 2
The function \(B ( z ) \) is not a meromorphic function over complex plane including \(z=0\).
Corollary 3
The function \(\mathbf{B} ( z ) \) is a meromorphic function at everywhere.

If \(a=\pi\), the pole is 0: that is, \(n ( a,\mathbf {B} ) =1\) where \(\mathbf{B}:=\mathbf{B} ( z ) \).

If \(a=2\pi\), the poles are \(2\pi i\), 0, \(2\pi i\): that is, \(n ( a,\mathbf{B} ) =3\).
Then we have the following corollary.
Corollary 4
Now we give the following theorem.
Theorem 4
Proof
Theorem 5
Proof
Theorem 6
3 Nevanlinna characteristic function of generating function of associated Euler numbers
Corollary 5
The function \(E(z)\) is a meromorphic function at everywhere.

If \(a=\pi\), the poles are \(\pi i\), πi: that is, \(n ( a,E ) =2\) where \(E:=E ( z ) \).

If \(a=3\pi\), the poles are \(3\pi i\), \(\pi i\), πi, \(3\pi i\): that is, \(n ( a,E ) =4\).
Then we have the following corollary.
Corollary 6
Now we give the following theorem.
Theorem 7
Proof
Because of Theorem 7 and Definition 3, we have the following corollary.
Corollary 7
Theorem 8
Proof
Theorem 9
4 Application
Thus we get the following theorem.
Theorem 10
The function \(\mathbf{B} ( z ) =\frac{1}{z}+\sum_{n=0}^{\infty}\frac{B_{n+1}}{n+1}\frac{z^{n}}{n!}\) is a normal function.
5 Conclusion and observation
By using this relation, one may derive easily the Nevanlinna characteristic function of the generating function of Genocchi numbers the same as the Nevanlinna characteristic function of the generating function of associated Euler numbers.
Declarations
Acknowledgements
The authors are very grateful to reviewers for their careful reading of our paper and for their valuable suggestions and comments, which have improved the paper’s presentation substantially.
Funding
The first author, Serkan Araci, is supported by the Research Fund of Hasan Kalyoncu University in 2018.
Authors’ contributions
All authors contributed equally to the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Biographies: Rolf Herman Nevanlinna. http://wwwhistory.mcs.standrews.ac.uk/Biographies/Nevanlinna.html
 Hayman, W.K.: Meromorphic Functions. Clarendon, Oxford (1964) MATHGoogle Scholar
 Lo, Y.: Value Distribution Theory. Springer, Berlin (1993) View ArticleMATHGoogle Scholar
 Li, P.: Entire solutions of certain type of differential equations. J. Math. Anal. Appl. 344, 253–259 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Li, P.: Entire solutions of certain type of differential equations II. J. Math. Anal. Appl. 375, 310–319 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Lu, X.L., Liao, L.W., Wang, J.: On meromorphic solutions of a certain type of nonlinear differential equations. Acta Math. Sin. Engl. Ser. 33(12), 1597–1608 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Wang, Y.M.: On normality of meromorphic functions with multiple zeros and sharing values. Turk. J. Math. 36, 263–271 (2012) MathSciNetMATHGoogle Scholar
 Wu, J., Wu, Z.: Characteristic function of a meromorphic function and its derivatives. Bull. Iran. Math. Soc. 39(6), 1117–1123 (2013) MathSciNetMATHGoogle Scholar
 Araci, S., Acikgoz, M., Sen, E.: Some new formulae for Genocchi numbers and polynomials involving Bernoulli and Euler polynomials. Int. J. Math. Math. Sci. 2014, Article ID 760613 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Kim, T.: Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(1), 23–28 (2010) MathSciNetMATHGoogle Scholar
 Srivastava, H.M.: Some generalizations and basic (or q) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 5(3), 390–444 (2011) MathSciNetGoogle Scholar