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 (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 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. 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. Expansion into a Newton series The expansion of the Bernoulli polynomials of the second kind into a Newton series reads 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 . See also Bernoulli polynomials Stirling polynomials Gregory coefficients Bernoulli numbers Difference polynomials Poly-Bernoulli number Mittag-Leffler polynomials References Mathematics Polynomials Number theory