Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas’ formulas.     

Eulerian quasisymmetric functions for the type B Coxeter group and other wreath product groups

Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler?s exponential generating function formula for the Eulerian numbers (Shareshian and Wachs, 2010 [17]). They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, which generalize the Shareshian–Wachs q-analog of Euler?s formula, formulas of Foata and Han, and a formula of Chow and Gessel.     

A method for proving polynomial enumeration formulas

We present an elementary method for proving enumeration formulas which are polynomials in certain parameters if others are fixed and factorize into distinct linear factors over Z. Roughly speaking the idea is to prove such formulas by “explaining” their zeros using an appropriate combinatorial extension of the objects under consideration to negative integer parameters. We apply this method to prove a new refinement of the Bender-Knuth (ex-)Conjecture, which easily implies the Bender-Knuth (ex-)Conjecture itself. This is probably the most elementary way to prove this result currently known. Furthermore we adapt our method to q-polynomials, which allows us to derive generating function results as well. Finally we use this method to give another proof for the enumeration of semistandard tableaux of a fixed shape which differs from our proof of the Bender-Knuth (ex-)Conjecture in that it is a multivariate application of our method.     

Généralisation d'identités de Carlitz,Howard et Lehmer

Divided differences provide an efficient method for computing with functions of several variables.In this note, we use them to generalize the Newton interpolation formula, and obtain an orthogonality relation (3.3). From this, we deduce two inversion formulas (3.4) and (3.8) involving two infinite sets of variables.The generating functions (1.1) to (1.4) of Carlitz and Howard are obtained by a mere specialization of variables in the preceding inversion formulas.As an illustration, we show how to recover several identities due to Carlitz and Lehmer, and we give a newq-analog of the generating series of the Howard numbers (formulas 4.22 and 4.23).

     

Integral geometry of unitary area measures

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the algebra associated to this co-product is described explicitly in terms of generators and relations. As a consequence, a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained.     

Counting involutory, unimodal, and alternating signed permutations

In this work we count the number of involutory, unimodal, and alternating elements of the group of signed permutations B_n, and the group of even-signed permutations D_n. Recurrence relations, generating functions, and explicit formulas of the enumerating sequences are given.     

Riordan arrays associated with Laurent series and generalized Sheffer-type groups

A relationship between a pair of Laurent series and Riordan arrays is formulated. In addition, a type of generalized Sheffer groups is defined by using Riordan arrays with respect to power series with non-zero coefficients. The isomorphism between a generalized Sheffer group and the group of the Riordan arrays associated with Laurent series is established. Furthermore, Appell, associated, Bell, and hitting-time subgroups of the groups are defined and discussed. A relationship between the generalized Sheffer groups with respect to different type of power series is presented. The equivalence of the defined Riordan array pairs and generalized Stirling number pairs is given. A type of inverse relations of various series is constructed by using pairs of Riordan arrays. Finally, several applications involving various arrays, polynomial sequences, special formulas and identities are also presented as illustrative examples.     

Representations of Central Matrix-Valued Carathéodory Functions in Both Nondegenerate and Degenerate Cases

We derive left and right quotient representations for central q × q matrix-valued Carathéodory functions. Moreover, we obtain recurrent formulas for the matrix polynomials involved in the quotient representations. These formulas are the starting point for getting recurrent formulas for those matrix polynomials which occur in the Arov-Krein resolvent matrix for the nondegenerate matricial Carathéodory problem.     

Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

The Padua points are a family of points on the square [−1, 1]² given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The L ^p convergence of the interpolation polynomials is also studied. S. De Marchi and M. Vianello were supported by the “ex-60%” funds of the University of Padua and by the INdAM GNCS (Italian National Group for Scientific Computing). Y. Xu was partially supported by NSF Grant DMS-0604056.     

Binomial Determinant Evaluations

By means of matrix decomposition and the partial fraction method, we establish several determinant evaluation formulas, which can be considered as generalizations of the Vandermonde and Cauchy determinants. Received February 24, 2005     

The Radon transform on hyperbolic space

The Radon transform that integrates a function in ⁿ , the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.     

Group inverses and Drazin inverses over Banach algebras

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.     

Positivity for Gaussian graphical models

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of non-zero subdeterminants.     

A symbolic operator approach to several summation formulas for power series II

Here expounded is a kind of symbolic operator method that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.     

Bijections and symmetries for the factorizations of the long cycle

We study the factorizations of the permutation (1,2,…,n)$(1,2,\dots ,n)$ into k factors of given cycle types. Using representation theory, Jackson obtained for each k   an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases k=2,3$k=2,3$ Schaeffer and Vassilieva gave a combinatorial proof of Jackson?s formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of (1,2,…,n)$(1,2,\dots ,n)$ into k factors for all k  . We thereby obtain refinements of Jackson?s formulas which extend the cases k=2,3$k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of “constellations”, which are graphs embedded in surfaces encoding the transitive factorizations of permutations.     

Translative integral formulae for convex bodies

Translative versions of the principal kinematic formula for quermassintegrals of convex bodies are studied. The translation integral is shown to be a sum of Crofton type integrals of mixed volumes. As corollaries new integral formulas for mixed volumes are obtained. For smooth centrally symmetric bodies the functionals occurring in the principal translative formula are expressed by measures on Grassmannians which are related to the generating measures of the bodies.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Bound states of a canonical system with a pseudo-exponential potential

Explicit formulas are given for the bound states (theL ²-eigenfunctions) and the corresponding eigenvalues of a self-adjoint operator defined by a canonical system with a pseudo-exponential potential. The formulas are expressed in terms of three matrices determining the potential. Both the half line and the full line case are considered.     

Hermite-Fejer type interpolation of higher order

The paper introduces Hermite-Fejér type (Hermite type) interpolation of higher order denoted by S _mn(f)(S* _mm(f)), and gives some basic properties including expression formulas, convergence relationship between S _mn(f) and H _mn(f) (Hermite-Fejér interpolation of higher order), and the saturation of S _mn(f). Supported by the Science Foundation of Shanxi Province for Returned Scholars.     

The multipoint Padé table and general recurrences for rational interpolation

We first review briefly the Newton-Padé approximation problem and the analogous problem with additional interpolation conditions at infinity, which we call multipoint Padé approximation problem. General recurrence formulas for the Newton-Padé table combine either two pairs of Newton-Padé forms or one such pair and a pair of multipoint Padé forms. We show that, likewise, certain general recurrences for the multipoint Padé table compose two pairs of multipoint Padé forms to get a new pair of multipoint Padé forms. We also discuss the possibility of superfast, i.e.,O(n log² n) algorithms for certain rational interpolation problems.     

Counting formulas for glued lattices

A tolerance relation of a lattice L, i.e., a reflexive and symmetric relation of L which is compatible with join and meet, is called glued if covering blocks of have nonempty intersection. For a lattice L with a glued tolerance relation we prove a formula counting the number of elements of L with exactly k lower (upper) covers. Moreover, we prove similar formulas for incidence structures and graphs and we give a new proof of Dilworth's covering theorem.