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 .
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:
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,
[6]
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
^Taylor, Brian D. (1998). "Difference Equations via the Classical Umbral Calculus". Mathematical Essays in honor of Gian-Carlo Rota. pp. 397–411. CiteSeerX 10.1.1.11.7516. doi:10.1007/978-1-4612-4108-9_21. ISBN 978-1-4612-8656-1.
^Di Nardo, E. (February 14, 2022). "A new approach to Sheppard's corrections". arXiv:1004.4989 [math.ST].
^"The classical umbral calculus: Sheffer sequences" (PDF). Lecture Notes of Seminario Interdisciplinare di Matematica. 8: 101–130. 2009.