Bernoulli Polynomials Of The Second Kind

The Bernoulli polynomials of the second kind , also known as the Fontana-Bessel polynomials,arXiv
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: $\frac= \sum_^\infty z^n \psi_n(x) ,\qquad , z, <1.$

The first five polynomials are:

:
\begin
\displaystyle \psi_0(x)=1 \\ mm\displaystyle \psi_1(x)=x+\frac12 \\ mm\displaystyle \psi_2(x)=\frac12x^2-\frac\\ mm\displaystyle \psi_3(x)=\frac16x^3-\frac14x^2+\frac\\ mm\displaystyle \psi_4(x)=\fracx^4-\frac16x^3+\frac16x^2 -\frac
\end

Some authors define these polynomials slightly differently

: $\frac= \sum_^\infty \frac \psi^*_n(x) ,\qquad , z, <1,$

so that

:$\psi^*_n(x)= \psi_n(x)\, n!$

and may also use a different notation for them (the most used alternative notation is ).

The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan, but their history may also be traced back to the much earlier works.

Integral representations

The Bernoulli polynomials of the second kind may be represented via these integrals

: $\psi_(x) = \int\limits_x^\! \binom\, du = \int\limits_0^1 \binom\, du$

as well as

:
\begin
\displaystyle \psi_(x)=\frac
\int\limits_0^\infty \frac \cdot\frac
,\qquad -1\leq x\leq n-1\, \\ mm\displaystyle \psi_(x)=\frac
\int\limits_^ \frac
\cdot\frac\, dv ,\qquad -1\leq x\leq n-1\,
\end

These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial.

Explicit formula

For an arbitrary , these polynomials may be computed explicitly via the following summation formula

:$\psi_(x) = \frac\sum_^ \frac x^ + G_,\qquad n=1,2,3,\ldots$

where are the signed Stirling numbers of the first kind and are the Gregory coefficients.

Recurrence formula

The Bernoulli polynomials of the second kind satisfy the recurrence relation

:$\psi_(x+1) - \psi_(x) = \psi_(x)$

or equivalently

:$\Delta\psi_(x) = \psi_(x)$

The repeated difference produces

:$\Delta^m\psi_(x) = \psi_(x)$

Symmetry property

The main property of the symmetry reads

:$\psi_(\tfrac12n-1+x) = (-1)^n\psi_(\tfrac12n-1-x)$

Some further properties and particular values

Some properties and particular values of these polynomials include
:
\begin
\displaystyle \psi_n(0)=G_n \\ mm\displaystyle \psi_n(1)=G_ + G_ \\ mm\displaystyle \psi_n(-1)= (-1)^ \sum_^n , G_m, = (-1)^n C_n\\ mm\displaystyle \psi_n(n-2)=-, G_n, \\ mm\displaystyle \psi_n(n-1)= (-1)^n \psi_n(-1) = 1- \sum_^n , G_m, \\ mm\displaystyle \psi_(n-1)=M_ \\ mm\displaystyle \psi_(n-1+y)=\psi_(n-1-y) \\ mm\displaystyle \psi_(n-\tfrac12+y)=-\psi_(n-\tfrac12-y) \\ mm\displaystyle \psi_(n-\tfrac12)=0
\end

where are the ''Cauchy numbers of the second kind'' and are the ''central difference coefficients''.

Expansion into a Newton series

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads

:$\psi_(x) = G_0 \binom + G_1 \binom + G_2 \binom + \ldots + G_n$

Some series involving the Bernoulli polynomials of the second kind

The digamma function may be expanded into a series with the Bernoulli polynomials of the second kind
in the following way

:$\Psi(v)=\ln(v+a) + \sum_^\infty\frac,\qquad \Re(v)>-a,$

and hence

$\gamma= -\ln(a+1) - \sum_^\infty\frac,\qquad \Re(a)>-1$

and

:$\gamma=\sum_^\infty\frac\Big\, \quad a>-1$

where is Euler's constant. Furthermore, we also have

: $\Psi(v)= \frac\left\,\qquad \Re(v)>-a,$

where is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these
polynomials as follows

: $\zeta(s,v)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^$

and

: $\zeta(s)= \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+1)^$

and also

: $\zeta(s) =1 + \frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+2)^$

The Bernoulli polynomials of the second kind are also involved in the following relationship

: $\big(v+a-\tfrac\big)\zeta(s,v) = -\frac + \zeta(s-1,v) + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom (k+v)^$

between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.

: $\gamma_m(v)=-\frac + \sum_^\infty (-1)^n \psi_(a) \sum_^ (-1)^k \binom\frac$

and

: $\gamma_m(v)=\frac \left\$

which are both valid for $\Re(a) > -1$ and $v\in\mathbb\setminus\!\$.

See also

* Bernoulli polynomials
* Stirling polynomials
* Gregory coefficients
* Bernoulli numbers
* Difference polynomials
* Poly-Bernoulli number
* Mittag-Leffler polynomials

References

{{Reflist, 2

Mathematics

Polynomials
Number theory