17 research outputs found
Faulhaber polynomials and reciprocal Bernoulli polynomials
About four centuries ago, Johann Faulhaber developed formulas for the power
sum in terms of . 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 the following properties
of the Faulhaber polynomials . The recurrences between , ,
and can be described by a certain differential operator. Furthermore,
we derive a recurrence formula for the coefficients of 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.Comment: 36 pages, 9 tables, 1 figure, revise
Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
The aim of this paper is to bring together the three objects in the title.
Recall that, given a Lie algebra , the Eulerian idempotent is a
canonical projection from the enveloping algebra to
. The Baker-Campbell-Hausdorff product and the Magnus expansion
can both be expressed in terms of the Eulerian idempotent, which makes it
interesting to establish explicit formulas for the latter. We show how to
reduce the computation of the Eulerian idempotent to the computation of a
logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The
problem of finding formulas for the pre-Lie logarithm, which is interesting in
its own right -- being related to operad theory, numerical analysis and
renormalization -- is addressed using techniques inspired by umbral calculus.
As a consequence of our analysis, we find formulas both for the Eulerian
idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.Comment: Preliminary version. Comments are welcome
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics
New Perspectives of Quantum Analogues
In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied
On Some Quadratic Algebras I : Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials
We study some combinatorial and algebraic properties of certain quadratic
algebras related with dynamical classical and classical Yang-Baxter equations.
One can find more details about the content of present paper in Extended
Abstract.Comment: Dedicated to the memory of Alain Lascoux (1944-2013). Preprint
RIMS-1817, 172 page
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Singularities, Supersymmetry and Combinatorial Reciprocity
This work illustrates a method to investigate certain smooth, codimension-two, real submanifolds of spheres of arbitrary odd dimension (with complements that fiber over the circle) using a novel supersymmetric quantum invariant. Algebraic (fibered) links, including Brieskorn-Pham homology spheres with exotic differentiable structure, are examples of said manifolds with a relative diffeomorphism-type that is determined by the corresponding (multivariate) Alexander polynomial.Engineering and Applied Science