Bell polynomials for sums and products with applications to derivatives of functions of trigonometric and hyperbolic functions
Abstract
The chain rule for the derivatives of a composite function gp o....o g1 can be succinctly expressed in terms of the Bell polynomials of the derivatives of the component functions, {B rk(gi)}. So, simple ways to compute Bell polynomials of sequences are of great interest. Bell polynomials of functions like power, exponential and log, whose derivatives are proportional to a geometric series, are particularly easy to obtain. We call such functions geometric. Here we give general expressions for the Bell polynomials of a sum of sequences or functions Brk(Σigi) in terms of the component Bell polynomials {Brk(gi)}- This gives an easy way to obtain Bell polynomials for functions which are the sum of geometric functions, such as sin, cos, sinh, cosh or polynomials. We also give Bell polynomials for products of functions in terms of the Bell polynomials of the component functions, that is Brk(Πigi) in terms of {B rk(gi)}.
