Bernoulli umbra

In Umbral calculus, the Bernoulli umbra

is an umbra, a formal symbol, defined by the relation

is the index-lowering operator,[1] also known as evaluation operator [2] and

are Bernoulli numbers, called moments of the umbra.

[3] A similar umbra, defined as

is also often used and sometimes called Bernoulli umbra as well.

They are related by equality

Along with the Euler umbra, Bernoulli umbra is one of the most important umbras.

In Levi-Civita field, Bernoulli umbras can be represented by elements with power series

ε

ε 24

ε

, with lowering index operator corresponding to taking the coefficient of

of the power series.

The numerators of the terms are given in OEIS A118050[4] and the denominators are in OEIS A118051.

are non-zero, the both are infinitely large numbers,

being infinitely close (but not equal, a bit smaller) to

being infinitely close (a bit smaller) to

In Hardy fields (which are generalizations of Levi-Civita field) umbra

corresponds to the germ at infinity of the function

corresponds to the germ at infinity of

is inverse digamma function.

Since Bernoulli polynomials is a generalization of Bernoulli numbers, exponentiation of Bernoulli umbra can be expressed via Bernoulli polynomials: where

is a real or complex number.

This can be further generalized using Hurwitz Zeta function: From the Riemann functional equation for Zeta function it follows that Since

are the only two members of the sequences

that differ, the following rule follows for any analytic function

: As a general rule, the following formula holds for any analytic function

: This allows to derive expressions for elementary functions of Bernoulli umbra.

Particularly, Particularly, Bernoulli umbra allows to establish relations between exponential, trigonometric and hyperbolic functions on one side and logarithms, inverse trigonometric and inverse hyperbolic functions on the other side in closed form: