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.
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.
Plot of the function , whose germ at positive infinity corresponds to .
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:
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