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Bernoulli polynomials of the second kind

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)).

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 representationsEdit

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]

Explicit formulaEdit

For an arbitrary n, these polynomials may be computed explicitly via the following summation formula[1][2][3]


where where s(n,l) are the signed Stirling numbers of the first kind and Gn are the Gregory coefficients.

Recurrence formulaEdit

The Bernoulli polynomials of the second kind satisfy the recurrence relation[1][2]


or equivalently


The repeated difference produces[1][2]


Symmetry propertyEdit

The main property of the symmetry reads[2][4]


Some further properties and particular valuesEdit

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]

Expansion into a Newton seriesEdit

The expansion of the Bernoulli polynomials of the second kind into a Newton series reads[1][2]


Some series involving the Bernoulli polynomials of the second kindEdit

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]




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 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]




which are both valid for   and  .

See alsoEdit


  1. ^ a b c d e f g h i Jordan, Charles (1928), "Sur des polynomes analogues aux polynomes de Bernoulli, et sur des formules de sommation analogues à celle de Maclaurin-Euler", Acta Sci. Math. (Szeged), 4: 130–150
  2. ^ a b c d e f g h i j Jordan, Charles (1965). The Calculus of Finite Differences (3rd Edition). Chelsea Publishing Company.
  3. ^ a b c d e f g h i j k l Blagouchine, Iaroslav V. (2018), "Three notes on Ser's and Hasse's representations for the zeta-functions" (PDF), Integers (Electronic Journal of Combinatorial Number Theory), 18A (#A3): 1–45 arXiv
  4. ^ a b Roman, S. (1984). The Umbral Calculus. New York: Academic Press.
  5. ^ Weisstein, Eric W. Bernoulli Polynomial of the Second Kind. From MathWorld--A Wolfram Web Resource.