In Umbral calculus, the Bernoulli umbra ${\displaystyle B_{-}}$ is an umbra, a formal symbol, defined by the relation ${\displaystyle \operatorname {eval} B_{-}^{n}=B_{n}^{-}}$, where ${\displaystyle \operatorname {eval} }$ is the index-lowering operator,^[1] also known as evaluation operator ^[2] and ${\displaystyle B_{n}^{-}}$ are Bernoulli numbers, called moments of the umbra.^[3] A similar umbra, defined as ${\displaystyle \operatorname {eval} B_{+}^{n}=B_{n}^{+}}$, where ${\displaystyle B_{1}^{+}=1/2}$ is also often used and sometimes called Bernoulli umbra as well. They are related by equality ${\displaystyle B_{+}=B_{-}+1}$. 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 ${\displaystyle B_{-}=\varepsilon ^{-1}-{\frac {1}{2}}-{\frac {\varepsilon }{24}}+{\frac {3\varepsilon ^{3}}{640}}-{\frac {1525\varepsilon ^{5}}{580608}}+\dotsb }$ and ${\displaystyle B_{+}=\varepsilon ^{-1}+{\frac {1}{2}}-{\frac {\varepsilon }{24}}+{\frac {3\varepsilon ^{3}}{640}}-{\frac {1525\varepsilon ^{5}}{580608}}+\dotsb }$, with lowering index operator corresponding to taking the coefficient of ${\displaystyle 1=\varepsilon ^{0}}$ 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 ${\displaystyle \varepsilon ^{-1}}$ are non-zero, the both are infinitely large numbers, ${\displaystyle B_{-}}$ being infinitely close (but not equal, a bit smaller) to ${\displaystyle \varepsilon ^{-1}-1/2}$ and ${\displaystyle B_{+}}$ being infinitely close (a bit smaller) to ${\displaystyle \varepsilon ^{-1}+1/2}$.

In Hardy fields (which are generalizations of Levi-Civita field) umbra ${\displaystyle B_{+}}$ corresponds to the germ at infinity of the function ${\displaystyle \psi ^{-1}(\ln x)}$ while ${\displaystyle B_{-}}$ corresponds to the germ at infinity of ${\displaystyle \psi ^{-1}(\ln x)-1}$, where ${\displaystyle \psi ^{-1}(x)}$ is inverse digamma function.

Plot of the function ${\displaystyle \psi ^{-1}(\ln(x))}$, whose germ at positive infinity corresponds to ${\displaystyle B_{+}}$.

Exponentiation

Since Bernoulli polynomials is a generalization of Bernoulli numbers, exponentiation of Bernoulli umbra can be expressed via Bernoulli polynomials:

${\displaystyle \operatorname {eval} (B_{-}+a)^{n}=B_{n}(a),}$

where ${\displaystyle a}$ is a real or complex number. This can be further generalized using Hurwitz Zeta function:

${\displaystyle \operatorname {eval} (B_{-}+a)^{p}=-p\zeta (1-p,a).}$

From the Riemann functional equation for Zeta function it follows that

${\displaystyle \operatorname {eval} \,B_{+}^{-p}=\operatorname {eval} {\frac {B_{+}^{p+1}2^{p}\pi ^{p+1}}{\sin(\pi p/2)\Gamma (p)(p+1)}}}$

Derivative rule

Since ${\displaystyle B_{1}^{+}=1/2}$ and ${\displaystyle B_{1}^{-}=-1/2}$ are the only two members of the sequences ${\displaystyle B_{n}^{+}}$ and ${\displaystyle B_{n}^{-}}$ that differ, the following rule follows for any analytic function ${\displaystyle f(x)}$:

${\displaystyle f'(x)=\operatorname {eval} (f(B_{+}+x)-f(B_{-}+x))=\operatorname {eval} \Delta f(B_{-}+x)}$

Elementary functions of Bernoulli umbra

As a general rule, the following formula holds for any analytic function ${\displaystyle f(x)}$:

${\displaystyle \operatorname {eval} f(B_{-}+x)={\frac {D}{e^{D}-1}}f(x).}$

This allows to derive expressions for elementary functions of Bernoulli umbra.

${\displaystyle \operatorname {eval} \cos(zB_{-})=\operatorname {eval} \cos(zB_{+})={\frac {z}{2}}\cot \left({\frac {z}{2}}\right)}$

${\displaystyle \operatorname {eval} \cosh(zB_{-})=\operatorname {eval} \cosh(zB_{+})={\frac {z}{2}}\coth \left({\frac {z}{2}}\right)}$

${\displaystyle \operatorname {eval} e^{zB_{-}}={\frac {z}{e^{z}-1}}}$

${\displaystyle \operatorname {eval} \ln(B_{-}+z)=\psi (z)}$

Particularly,

${\displaystyle \operatorname {eval} \ln B_{+}=-\gamma }$ ^[6]

${\displaystyle \operatorname {eval} {\frac {1}{\pi }}\ln \left({\frac {B_{+}-{\frac {z}{\pi }}}{B_{-}+{\frac {z}{\pi }}}}\right)=\cot z}$

${\displaystyle \operatorname {eval} {\frac {1}{\pi }}\ln \left({\frac {B_{-}+1/2+{\frac {z}{\pi }}}{B_{-}+1/2-{\frac {z}{\pi }}}}\right)=\tan z}$

${\displaystyle \operatorname {eval} \cos(aB_{-}+x)={\frac {a}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-x\right)}$

${\displaystyle \operatorname {eval} \sin(aB_{-}+x)={\frac {a}{2}}\cot \left({\frac {a}{2}}\right)\sin x-{\frac {a}{2}}\cos x}$

Particularly,

${\displaystyle \operatorname {eval} \sin B_{-}=-1/2}$,

${\displaystyle \operatorname {eval} \sin B_{+}=1/2}$,

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:

${\displaystyle \operatorname {eval} \left(\cosh \left(2xB_{\pm }\right)-1\right)=\operatorname {eval} {\frac {x}{\pi }}\operatorname {artanh} \left({\frac {x}{\pi B_{\pm }}}\right)=\operatorname {eval} {\frac {x}{\pi }}\operatorname {arcoth} \left({\frac {\pi B_{\pm }}{x}}\right)=x\coth(x)-1}$

${\displaystyle \operatorname {eval} {\frac {z}{2\pi }}\ln \left({\frac {B_{+}-{\frac {z}{2\pi }}}{B_{-}+{\frac {z}{2\pi }}}}\right)=\operatorname {eval} \cos(zB_{-})=\operatorname {eval} \cos(zB_{+})={\frac {z}{2}}\cot \left({\frac {z}{2}}\right)}$

References