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Bernoulli umbra
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In Umbral calculus, the Bernoulli umbra is an umbra, a formal symbol, defined by the relation , where 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 , 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.
![]() | This article may be too technical for most readers to understand. (April 2022) |
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[4] and the denominators are in OEIS A118051.[5] 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.

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Exponentiation
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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
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Derivative rule
Since and are the only two members of the sequences and that differ, the following rule follows for any analytic function :
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Elementary functions of Bernoulli umbra
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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,
- ,
- ,
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Relations between exponential and logarithmic functions
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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:
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References
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