Central differences, Euler numbers and symbolic methods

TL;DR

This paper explores the connections between central differences, Euler numbers, and symbolic calculus, providing new sum rules, a symbolic framework, and computational advantages for these mathematical objects.

Contribution

It introduces a general symbolic treatment of central difference calculus, deriving new sum rules and expanding the computational tools for Euler numbers and related functions.

Findings

Derived sum rules for central differentials and Euler numbers.

Presented a symbolic framework for central difference calculus.

Obtained expansions of powers of inverse sinh with minimal effort.

Abstract

I relate some coefficients encountered when computing the functional determinants on spheres to the central differentials of nothing. In doing this I use some historic works, in particular transcribing the elegant symbolic formalism of Jeffery (1861) into central difference form which has computational advantages for Euler numbers, as discovered by Shovelton (1915). I derive sum rules for these, and for the central differentials, the proof of which involves an interesting expression for powers of sech x as multiple derivatives. I present a more general, symbolic treatment of central difference calculus which allows known, and unknown, things to be obtained in an elegant and compact fashion gaining, at no cost, the expansion of the powers of the inverse sinh, a basic central function. Systematic use is made of the operator 2 asinh(D/2). Umbral calculus is employed to compress the operator formalism. For example the orthogonality/completeness of the factorial numbers, of the first and second kinds, translates, umbrally, to T(t(x))=x. The classic expansions of multiple angle cosh and sinh in terms of powers of sinh are thence obtained with minimal effort.