Simplification of coefficients in differential equations associated with higher order Frobenius--Euler numbers

Abstract

In the paper, by virtue of the Fa\`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius--Euler numbers.