Bernoulli umbra

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.

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

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

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 ${\displaystyle f(x)}$:

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

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,

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

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

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:

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

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