Finding non-linear differential equations and certain identities for the Bernoulli–Euler and Bernoulli–Genocchi numbers

Abstract

In this research paper, the authors derived the non-linear differential equations for certain hybrid special polynomials related to the Bernoulli polynomials. The families of non-linear differential equations arising from the generating functions of the Bernoulli–Euler and Bernoulli–Genocchi polynomials are derived. Further, these non-linear 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.