Bernoulli umbra
In Umbral calculus, the Bernoulli umbra  is an umbra, a formal symbol, defined by the relation , where  is the index-lowering operator, also known as evaluation operator  and  are Bernoulli numbers, called moments of the umbra. A similar umbra, defined as , where  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  and , with lowering index operator corresponding to taking the coefficient of  of the power series. The numerators of the terms are given in OEIS A118050 and the denominators are in OEIS A118051. Since the coefficients of  are non-zero, the both are infinitely large numbers,  being infinitely close (but not equal, a bit smaller) to  and  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  while  corresponds to the germ at infinity of , where  is inverse digamma function. Exponentiation

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

Derivative rule

Since  and  are the only two members of the sequences  and  that differ, the following rule follows for any analytic function :

Elementary functions of Bernoulli umbra

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,

, 
,

Relations between exponential and logarithmic functions

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:

References 

Polynomials
Finite differences
Combinatorics
Factorial and binomial topics