Bernoulli umbra

In Umbral calculus, the Bernoulli umbra $B_-$ is an umbra, a formal symbol, defined by the relation $\operatorname{eval}B_-^n=B^-_n$ , where $\operatorname{eval}$ is the index-lowering operator,{{Cite book|citeseerx = 10.1.1.11.7516|chapter = Difference Equations via the Classical Umbral Calculus|year = 1998|pages = 397–411|doi=10.1007/978-1-4612-4108-9_21|isbn=978-1-4612-8656-1|title= Mathematical Essays in honor of Gian-Carlo Rota|last1= Taylor|first1= Brian D.}} also known as evaluation operator {{cite arXiv|last1=Di Nardo |first1=E. |title=A new approach to Sheppard's corrections |date=February 14, 2022 |class=math.ST |eprint= 1004.4989 }} and $B^-_n$ are Bernoulli numbers, called moments of the umbra.{{cite journal |title=The classical umbral calculus: Sheffer sequences |journal= Lecture Notes of Seminario Interdisciplinare di Matematica |date= 2009 |volume= 8 |pages = 101–130 |url=http://math.fau.edu/Niederhausen/HTML/Papers/The%20classical%20umbral%20calculus%20dinardo_09.pdf}} A similar umbra, defined as $\operatorname{eval}B_+^n=B^+_n$ , where $B^+_1=1/2$ is also often used and sometimes called Bernoulli umbra as well. They are related by equality $B_+=B_-+1$ . 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 $B_-= \varepsilon^{-1} -\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$ and $B_+= \varepsilon^{-1} +\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$ , with lowering index operator corresponding to taking the coefficient of $1=\varepsilon^0$ of the power series. The numerators of the terms are given in OEIS A118050{{Cite OEIS|1=A118050|mode=cs2}} and the denominators are in OEIS A118051.{{Cite OEIS|1=A118051|mode=cs2}} Since the coefficients of $\varepsilon^{-1}$ are non-zero, the both are infinitely large numbers, $B_-$ being infinitely close (but not equal, a bit smaller) to $\varepsilon^{-1}-1/2$ and $B_+$ being infinitely close (a bit smaller) to $\varepsilon^{-1}+1/2$ .

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

File:Bumbra.png

Exponentiation

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

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

where $a$ is a real or complex number.

This can be further generalized using Hurwitz Zeta function:

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

From the Riemann functional equation for Zeta function it follows that

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

Derivative rule

Since $B^+_1=1/2$ and $B^-_1=-1/2$ are the only two members of the sequences $B^+_n$ and $B^-_n$ that differ, the following rule follows for any analytic function $f(x)$ :

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

Elementary functions of Bernoulli umbra

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

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

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

: $\operatorname{eval} \cos (z B_-)=\operatorname{eval} \cos (z B_+)=\frac z2 \cot \left(\frac z2\right)$

: $\operatorname{eval} \cosh (z B_-)=\operatorname{eval} \cosh (z B_+)=\frac z2 \coth \left(\frac z2\right)$

: $\operatorname{eval} e^{z B_-}=\frac{z}{e^{z}-1}$

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

Particularly,

: $\operatorname{eval}\ln B_+=-\gamma$ {{Cite arXiv |eprint = 1011.3352|last1 = Yu|first1 = Yiping|title = Bernoulli Operator and Riemann's Zeta Function|year = 2010|class = math.NT}}

: $\operatorname{eval}\frac1{\pi }\ln \left(\frac{ B _+-\frac{z}{\pi }}{ B _-+\frac{z}{\pi }}\right)=\cot z$

: $\operatorname{eval} \frac1\pi\ln \left(\frac{B _-+1/2 +\frac{z}{\pi }}{B _-+1/2 -\frac{z}{\pi }}\right)=\tan z$

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

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

Particularly,

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

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

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:

: $\operatorname{eval}\left(\cosh \left(2 x B _\pm\right)-1\right)=\operatorname{eval}\frac{x}{\pi} \operatorname{artanh}\left(\frac{x}{\pi B _\pm}\right)=\operatorname{eval}\frac{x}{\pi} \operatorname{arcoth}\left(\frac{\pi B _\pm}{x}\right)=x \coth (x)-1$

: $\operatorname{eval}\frac{z}{2\pi }\ln \left(\frac{ B _+-\frac{z}{2\pi }}{ B _-+\frac{z}{2\pi }}\right)=\operatorname{eval} \cos (z B_-)=\operatorname{eval} \cos (z B_+)=\frac z2 \cot \left(\frac z2\right)$

References

Category:Polynomials

Category:Finite differences

Category:Combinatorics

Category:Factorial and binomial topics