Bernoulli polynomials of the second kind

The Bernoulli polynomials of the second kind, also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating function:
The first five polynomials are:
Some authors define these polynomials slightly differently
so that
and may also use a different notation for them. 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, 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
as well as
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
where are the signed Stirling numbers of the first kind and are the Gregory coefficients.
The expansion of the Bernoulli polynomials of the second kind into a Newton series reads
It can be shown using the second integral representation and Vandermonde's identity.

Recurrence formula

The Bernoulli polynomials of the second kind satisfy the recurrence relation
or equivalently
The repeated difference produces

Symmetry property

The main property of the symmetry reads

Some further properties and particular values

Some properties and particular values of these polynomials include
where are the Cauchy numbers of the second kind and are the central difference coefficients.

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
and hence
and
where is Euler's constant. Furthermore, we also have
where is the gamma function. The Hurwitz and Riemann zeta functions may be expanded into these polynomials as follows
and
and also
The Bernoulli polynomials of the second kind are also involved in the following relationship
between the zeta functions, as well as in various formulas for the Stieltjes constants, e.g.
and
which are both valid for and.