Polynomial sequence
The Bernoulli polynomials of the second kind[1][2] ψn(x), also known as the Fontana–Bessel polynomials,[3] are the polynomials defined by the following generating function:
The first five polynomials are:
Some authors define these polynomials slightly differently[4][5]
so that
and may also use a different notation for them (the most used alternative notation is bn(x)). Under this convention, the polynomials form a Sheffer sequence.
The Bernoulli polynomials of the second kind were largely studied by the Hungarian mathematician Charles Jordan,[1][2] but their history may also be traced back to the much earlier works.[3]
Integral representations
[edit]
The Bernoulli polynomials of the second kind may be represented via these integrals[1][2]
as well as[3]
These polynomials are, therefore, up to a constant, the antiderivative of the binomial coefficient and also that of the falling factorial.[1][2][3]
For an arbitrary n, these polynomials may be computed explicitly via the following summation formula[1][2][3]
where s(n,l) are the signed Stirling numbers of the first kind and Gn are the Gregory coefficients.
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads[1][2]
It can be shown using the second integral representation and Vandermonde's identity.
The Bernoulli polynomials of the second kind satisfy the recurrence relation[1][2]
or equivalently
The repeated difference produces[1][2]
The main property of the symmetry reads[2][4]
Some further properties and particular values
[edit]
Some properties and particular values of these polynomials include
where Cn are the Cauchy numbers of the second kind and Mn are the central difference coefficients.[1][2][3]
Some series involving the Bernoulli polynomials of the second kind
[edit]
The digamma function Ψ(x) may be expanded into a series with the Bernoulli polynomials of the second kind
in the following way[3]
and hence[3]
and
where γ is Euler's constant. Furthermore, we also have[3]
where Γ(x) is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows[3]
and
and also
The Bernoulli polynomials of the second kind are also involved in the following relationship[3]
between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.[3]
and
which are both valid for
and
.