Bernoulli umbra

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

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

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

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

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

From the Riemann functional equation for Zeta function it follows that

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

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

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

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

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

Particularly,

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

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

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

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