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)). 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

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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 formula

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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.

Recurrence formula

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

The repeated difference produces[1][2]  

Symmetry property

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

Some further properties and particular values

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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

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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  .

See also

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References

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  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: The 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.

Mathematics

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