Finding nonlinear differential equations and certain identities for the Bernoulli–Euler and Bernoulli–Genocchi numbers
Abstract
In this research paper, the authors derived the nonlinear differential equations for certain hybrid special polynomials related to the Bernoulli polynomials. The families of nonlinear differential equations arising from the generating functions of the Bernoulli–Euler and Bernoulli–Genocchi polynomials are derived. Further, these nonlinear differential equations are used to derive certain identities and formulas for the Bernoulli–Euler and Bernoulli–Genocchi numbers. However, to provided an exception, a linear differential equation is derived from the generating function of the Genocchi–Euler polynomials.
Keywords
Bernoulli–Euler numbers Bernoulli–Genocchi numbers Genocchi–Euler numbers Nonlinear differential equationsMathematics Subject Classification
33E30 33E201 Introduction
There has been considerable progress during the recent past on mathematical techniques for studying differential equations that arise in science and engineering. The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. The problems arising in different areas of science and engineering are usually expressed in terms of differential equations, which in most of the cases have special functions as their solutions. The differential equations and recurrence relations for the Appell polynomials are studied by several authors, see for example [2, 11]. Recently, the recurrence relations and differential equations for certain hybrid special polynomials related to the Appell sequences are established, see for example [4, 15].
During the past 3 decades, the development of nonlinear analysis, dynamical systems and their applications to science and engineering has stimulated renewed enthusiasm for the theory of differential equations. Differential equations play an important role in modeling virtually every physical, technical or biological process ranging from celestial motion to bridge design and to interactions between neurons. Many fundamental laws of physics and chemistry can be formulated in the form of differential equations.
Nonlinear differential equations have been extensively used to mathematically model many interesting and important phenomena that are observed in many areas of science and technology. They are inspired by problems which arise in diverse fields such as economics, fluid mechanics, physics, differential geometry, engineering, control theory, material science and quantum mechanics. Recently, the linear and nonlinear differential equations from the generating functions of special polynomials and numbers, see for example [5, 6, 7, 8, 9, 10]. Kim in [5] initiated a remarkable idea of using nonlinear differential equations as a method of obtaining new identities for special polynomials and numbers. This method turned out to be very useful and it can be applied to many interesting special polynomials and numbers. The recurrence relations and associated differential equations are of fundamental importance in wide variety of fields of pure and applied mathematics, physics and engineering.
A nonlinear differential equation is generally more difficult to solve than linear equations. It is common that a nonlinear equation is approximated as linear equation for many practical problems, either in analytical or numerical form. The nonlinear nature of the problem is then approximated as series of linear differential equations by simple increment or with correction deviation from the nonlinear behaviour. This approach is adopted for the solution of many nonlinear engineering problems. Without such procedure most of the nonlinear differential equations can not be solved. There are very few methods of solving nonlinear differential equations exactly, those that are known typically depend on the equations having particular symmetries.
Certain members belonging to the Appell family
S. no.  Name of the polynomials and related numbers  A(t)  Generating function  Series definition 

I.  Bernoulli polynomials and numbers [14]  \(\left( \frac{t}{e^{t}1}\right)\)  \(\begin{array}{ll} \left( \frac{t}{e^{t}1}\right) e^{xt}=\sum \nolimits _{n=0}^\infty B_{n}(x)\frac{t^n}{n!}\\ \left( \frac{t}{e^{t}1}\right) =\sum \nolimits _{ n=0}^\infty B_{n}\frac{t^n}{n!}\\ B_{n}:=B_{n}(0) \end{array}\)  \(B_{n}(x)=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} B_{k} x^{nk}\) 
II.  Euler polynomials and numbers [14]  \(\left( \frac{2}{e^{t}+1}\right)\)  \(\begin{array}{ll}\left( \frac{2}{e^{t}+1}\right) e^{xt}=\sum \nolimits _{n=0}^\infty E_{n}(x)\frac{t^n}{n!}\\ \frac{2}{e^{t}+1}=\sum \nolimits _{n=0}^{\infty }E_{n}\frac{t^{n}}{n!} \\ E_{n}:= E_{n}(0) \end{array}\)  \(E_{n}(x)=\sum \nolimits _{k=0}^{n}{n\atopwithdelims ()k} {E_{k}}\,x^{nk}\) 
III.  Genocchi polynomials and numbers [13]  \(\left( \frac{2t}{e^{t}+1}\right)\)  \(\begin{array}{ll}\left( \frac{2t}{e^{t}+1}\right) e^{xt}=\sum \nolimits _{n=0}^\infty G_{n}(x)\frac{t^n}{n!}\\ \frac{2t}{e^{t}+1}=\sum \nolimits _{n=0}^{\infty }G_{n}\frac{t^{n}}{n!} \\ G_{n}:=G_n(0) \end{array}\)  \(G_{n}(x)=\sum \nolimits _{k=0}^{n}{n\atopwithdelims ()k}G_k x^{nk}\) 
The Bernoulli, Euler and Genocchi numbers have deep connections with number theory and occur in combinatorics. The Bernoulli numbers enter in many mathematical formulas, such as the Taylor expansion in a neighborhood of the origin of the trigonometric and hyperbolic tangent and cotangent functions and the sums of powers of natural numbers. The Euler numbers are strictly connected with the Bernoulli ones and enter in the Taylor expansion in a neighborhood of the origin of the trigonometric and hyperbolic secant functions. The Genocchi numbers are employed in wide range of applications in number theory, combinatorics, numerical analysis and other fields of applied mathematics. The Genocchi numbers are known to count a large variety of combinatorial objects such as sets of permutations.
Certain members belonging to the 2iterated Appell family
S. no.  Name of the hybrid polynomials and related numbers  \(A(t); \,{\mathcal {G}}(t)\)  Generating function  Series definition 

I.  Bernoulli–Euler polynomials and numbers  \(\left( \frac{t}{e^{t}1}\right) ;\,\left( \frac{2}{e^{t}+1}\right)\)  \(\begin{array}{ll}\left( \frac{2t}{e^{2t}1}\right) e^{xt}=\sum \nolimits _{n=0}^\infty {_B}E_{n}(x)\frac{t^n}{n!}\\ \left( \frac{2t}{e^{2t}1}\right) =\sum \nolimits _{n=0}^\infty {_B}E_{n}\frac{t^n}{n!}\\ {_B}E_{n}:={_B}E_{n}(0) \end{array}\)  \(\begin{array}{ll} {_B}E_{n}(x)=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} B_{k} E_{nk}(x)\\ {_B}E_{n}=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} B_{k} E_{nk} \end{array}\) 
II.  Bernoulli–Genocchi polynomials and numbers  \(\left( \frac{t}{e^{t}1}\right) ;\,\left( \frac{2t}{e^{t}+1}\right)\)  \(\begin{array}{ll}\left( \frac{2t^2}{e^{2t}1}\right) e^{xt}=\sum \nolimits _{n=0}^\infty {_B}G_{n}(x)\frac{t^n}{n!}\\ \left( \frac{2t^2}{e^{2t}1}\right) =\sum \nolimits _{n=0}^\infty {_B}G_{n}\frac{t^n}{n!}\\ {_B}G_{n}:={_B}G_{n}(0) \end{array}\)  \(\begin{array}{ll}{_B}G_{n}(x)=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} B_{k} G_{nk}(x)\\ {_B}G_{n}=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} B_{k} G_{nk}\end{array}\) 
III.  Genocchi–Euler polynomials and numbers  \(\left( \frac{2t}{e^{t}+1}\right) ;\,\left( \frac{2}{e^{t}+1}\right)\)  \(\begin{array}{ll}\left( \frac{4t}{(e^{t}+1)^2}\right) e^{xt}=\sum \nolimits _{n=0}^\infty {_G}E_{n}(x)\frac{t^n}{n!}\\ \left( \frac{4t}{(e^{t}+1)^2}\right) =\sum \nolimits _{n=0}^\infty {_G}E_{n}\frac{t^n}{n!}\\ {_G}E_{n}:={_G}E_{n}(0) \end{array}\)  \(\begin{array}{ll}{_G}E_{n}(x)=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} G_{k} E_{nk}(x)\\ {_G}E_{n}=\sum \nolimits _{k=0}^n {n \atopwithdelims ()k} G_{k} E_{nk} \end{array}\) 
The set of Appell sequences is closed under the operation of umbral composition of polynomial sequences. Under this operation the set of Appell sequences is an abelian group. Since the generating function of the 2IAP is of the form \(A^{\star }(t)e^{xt}\), with \(A^{\star }(t)\) as the product of two different functions of t. Therefore, the set of all 2IAP sequences also forms an abelian group under the operation of umbral composition. As a consequence of this fact, the Bernoulli–Euler polynomials are equivalent to the Euler–Bernoulli polynomials \(_EB_n(x)\), i.e., \(_BE_n(x)\equiv {_E}B_n(x)\). Similarly, the Bernoulli–Genocchi polynomials are equivalent to the Genocchi–Bernoulli polynomials \(_GB_n(x)\), i.e., \(_BG_n(x)\equiv {_G}B_n(x)\) and Genocchi–Euler polynomials are equivalent to the Euler–Genocchi polynomials \(_EG_n(x)\), i.e., \(_GE_n(x)\equiv {_E}G_n(x)\).
The content of this article are motivated by the work under progress in the direction of obtaining linear and nonlinear differential equations involving special polynomials. In this article, the families of nonlinear differential equations related to the generating functions of the Bernoulli–Euler and Bernoulli–Genocchi polynomials are constructed. Certain identities for the hybrid special polynomials are established using these nonlinear differential equations. To provide an exception, a family of linear differential equations is also derived from the generating function of the Genocchi–Euler polynomials.
2 Methodologies
We derive the families of nonlinear differential equations from the generating functions of the Bernoulli–Euler polynomials \(_BE_n(x)\) and Bernoulli–Genocchi polynomials \({_B}G_n(x)\).
Theorem 2.1
Proof
The following corollary is an immediate consequence of Theorem 2.1.
Corollary 2.1
Proof
Setting \(H^{(j)}(t,x) = H^{(j)}(t) e^{xt}\) and \(H^N(t, x) = \underbrace{H(t) \times H(t) \times \cdots \times H(t)}_{Ntimes} e^{xt}\) and multiplying both sides of Eq. (2.1) by \(e^{xt}\), we get assertion (2.16). \(\square\)
Theorem 2.2
Proof
The following corollary is an immediate consequence of Theorem 2.2.
Corollary 2.2
Proof
Setting \(G^{(N)}(t,x) = G^{(N)}(t) e^{xt}\) and \(G^i(t, x) = \underbrace{G(t) \times G(t) \times \cdots \times G(t)}_{itimes} e^{xt}\) and multiplying both sides of Eq. (2.17) by \(e^{xt}\), we get assertion (2.39). \(\square\)
In the next section, we establish certain identities arising from the nonlinear differential equations obtained above.
3 Results and summary
In view of Theorem 2.1, we derive the identities related to the Bernoulli–Euler numbers \({_B}E_n\) by proving the following result:
Theorem 3.1
Proof
Corollary 3.1
Proof
Next, in view of Theorem 2.2, we derive the identities related to the Bernoulli–Genocchi numbers \({_B}G_n\) by proving the following result:
Theorem 3.2
Proof
Corollary 3.2
Proof
Following the same lines of proof as in Corollary 3.1, assertion (3.13) is proved. \(\square\)
In the next section, a family of linear differential equation from the generating function of the Genocchi–Euler polynomials \({_G}E_n(x)\) is derived.
4 Conclusions
In the previous section, the nonlinear differential equations from the generating functions of the hybrid polynomials related to the Bernoulli, Euler and Genocchi polynomials are considered. However, it is not necessary that every hybrid family yields a nonlinear differential equation. To give an example, we consider the Genocchi–Euler polynomials \({_G}E_n(x)\).
Theorem 4.1

The Bernoulli–Euler and Bernoulli–Genocchi polynomials belong to the extended Appell class and form an abelian group under the operation of umbral composition.

In this paper, the nonlinear differential equations for the hybrid members belonging to this extended class are derived.

The differential equations related to the generating functions of the hybrid special polynomials derived in this article are important from the point of view of their applications in various fields of science.

It has been demonstrated that it is a captivating idea to use differential equations arising from the generating functions of the hybrid special polynomials to derive enthralling identities related to special polynomials and numbers.

The approach presented in this article is general and can be extended to other families of hybrid special polynomials.
Notes
Acknowledgements
This work has been done under Senior Research Fellowship [Award letter No. F./201415/NFO201415OBCUTT24168/(SAIII/Website)] awarded to the second author by the University Grants Commission, Government of India, New Delhi.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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