Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Faulhaber polynomials and reciprocal Bernoulli polynomials   About four centuries ago, Johann Faulhaber developed formulas for the power
sum $1^n + 2^n + \cdots + m^n$ in terms of $m(m+1)/2$. The resulting
polynomials are called the Faulhaber polynomials. We first give a short survey
of Faulhaber's work and discuss the results of Jacobi (1834) and the less known
ones of Schr\"oder (1867), which already imply some results published
afterwards. We then show for suitable odd integers $n$ the following properties
of the Faulhaber polynomials $F_n$. The recurrences between $F_n$, $F_{n-1}$,
and $F_{n-2}$ can be described by a certain differential operator. Furthermore,
we derive a recurrence formula for the coefficients of $F_n$ that is the
complement of a formula of Gessel and Viennot (1989). As a main result, we show
that these coefficients can be expressed and computed in different ways by
derivatives of generalized reciprocal Bernoulli polynomials, whose values can
be also interpreted as central coefficients. This new approach finally leads to
a simplified representation of the Faulhaber polynomials. As an application, we
obtain some recurrences of the Bernoulli numbers, which are induced by symmetry
properties.

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The categories of this paper are: econ
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: alg-geom