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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.

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

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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

Derivative rule

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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

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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,

[6]

Particularly,

,
,

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:

References

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  1. ^ 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.
  2. ^ Di Nardo, E. (February 14, 2022). "A new approach to Sheppard's corrections". arXiv:1004.4989 [math.ST].
  3. ^ "The classical umbral calculus: Sheffer sequences" (PDF). Lecture Notes of Seminario Interdisciplinare di Matematica. 8: 101–130. 2009.
  4. ^ Sloane, N. J. A. (ed.), "Sequence A118050", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  5. ^ Sloane, N. J. A. (ed.), "Sequence A118051", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  6. ^ Yu, Yiping (2010). "Bernoulli Operator and Riemann's Zeta Function". arXiv:1011.3352 [math.NT].