scholarly journals A note on moments of derivatives of characteristic polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Paul-Olivier Dehaye
Keyword(s):  
Hypergeometric Functions ◽  
Simple Technique ◽  
Schur Functions ◽  
Unitary Groups ◽  
Characteristic Polynomials ◽  
Nous Obtenons ◽  
Binomial Theorem ◽  
Taylor Expansions ◽  
International Audience ◽  
Derivatives Of

International audience We present a simple technique to compute moments of derivatives of unitary characteristic polynomials. The first part of the technique relies on an idea of Bump and Gamburd: it uses orthonormality of Schur functions over unitary groups to compute matrix averages of characteristic polynomials. In order to consider derivatives of those polynomials, we here need the added strength of the Generalized Binomial Theorem of Okounkov and Olshanski. This result is very natural as it provides coefficients for the Taylor expansions of Schur functions, in terms of shifted Schur functions. The answer is finally given as a sum over partitions of functions of the contents. One can also obtain alternative expressions involving hypergeometric functions of matrix arguments. Nous introduisons une nouvelle technique, en deux parties, pour calculer les moments de dérivées de polynômes caractéristiques. La première étape repose sur une idée de Bump et Gamburd et utilise l'orthonormalité des fonctions de Schur sur les groupes unitaires pour calculer des moyennes de polynômes caractéristiques de matrices aléatoires. La deuxième étape, qui est nécessaire pour passer aux dérivées, utilise une généralisation du théorème binomial due à Okounkov et Olshanski. Ce théorème livre les coefficients des séries de Taylor pour les fonctions de Schur sous la forme de "shifted Schur functions''. La réponse finale est donnée sous forme de somme sur les partitions de fonctions des contenus. Nous obtenons aussi d'autres expressions en terme de fonctions hypergéométriques d'argument matriciel.

scholarly journals The down operator and expansions of near rectangular k-Schur functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Chris Berg ◽  
Franco Saliola ◽  
Luis Serrano
Keyword(s):  
Weyl Group ◽  
Schur Functions ◽  
Nous Obtenons ◽  
Combinatorial Interpretation ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience

International audience We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients. Nous montrons que l’opérateur ``down'', défini par Lam et Shimozono sur le groupe de Weyl affine, induit une dérivation de la sous-algèbre affine de Fomin-Stanley de l'algèbre affine de nilCoxeter. Nous employons cette dérivation pour vérifier une conjecture de Berg, Bergeron, Pon et Zabrocki sur l'expansion des k-fonctions de Schur indexées par les partitions qui sont ``presque rectangles''. Par conséquent, nous obtenons une interprétation combinatoire des k-coefficients de Littlewood–Richardson correspondants.

scholarly journals The ABC's of affine Grassmannians and Hall-Littlewood polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Schur Functions ◽  
Type A ◽  
Combinatorial Formula ◽  
Transition Matrices ◽  
Nous Obtenons ◽  
Affine Grassmannians ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Affine Type

International audience We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.

scholarly journals Row-strict quasisymmetric Schur functions

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sarah K Mason ◽  
Jeffrey Remmel
Keyword(s):  
Schur Functions ◽  
Schur Function ◽  
Nous Obtenons ◽  
Schur Polynomials ◽  
Quasisymmetric Functions ◽  
International Audience ◽  
Quasisymmetric Schur Functions ◽  
The Relationship

International audience Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships. Haglund, Luoto, Mason, et van Willigenburg ont introduit une base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "column-strict'' (ordre strict sur les colonnes). Nous introduisons une nouvelle base pour les fonctions quasi-symétriques appelée $\textit{base des fonctions de Schur quasi-symétriques "row-strict''}$, qui sont construites en remplissant des diagrammes de compositions, d'une manière très semblable à la construction des fonctions de Schur à partir des tableaux "row-strict'' (ordre strict sur les lignes). Nous décrivons la relation entre cette nouvelle base et d'autres bases connues pour les fonctions quasi-symétriques, ainsi que ses relations avec les polynômes de Schur. Nous obtenons un raffinement de l'opérateur oméga comme conséquence de ces relations.

scholarly journals General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1102
Author(s):  
Yashoverdhan Vyas ◽  
Hari M. Srivastava ◽  
Shivani Pathak ◽  
Kalpana Fatawat
Keyword(s):  
Hypergeometric Functions ◽  
Theory Theory ◽  
Binomial Theorem ◽  
Generalized Hypergeometric Functions ◽  
Quantum Calculus ◽  
The Third ◽  
Summation Formulas ◽  
Symmetric Quantum ◽  
Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Application of Maple on Solving the Differential Problem of Rational Functions

2013 ◽  
Vol 479-480 ◽  
pp. 855-860
Author(s):  
Chii Huei Yu
Keyword(s):  
Problem Solving ◽  
Research Methods ◽  
Research Method ◽  
Rational Functions ◽  
The Other ◽  
Mathematical Software ◽  
Differential Problem ◽  
Binomial Theorem ◽  
Other Hand ◽  
Derivatives Of

This paper uses the mathematical software Maple as the auxiliary tool to study the differential problem of four types of rational functions. We can obtain the closed forms of any order derivatives of these rational functions by using binomial theorem. On the other hand, we propose four examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

scholarly journals Enumerating alternating tree families

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Nous Obtenons ◽  
Asymptotic Results ◽  
Root Node ◽  
Ordered Trees ◽  
Exponential Generating Function ◽  
Random Node ◽  
Enumeration Problems ◽  
International Audience ◽  
Nous Examinons ◽  
Labelled Trees

International audience We study two enumeration problems for $\textit{up-down alternating trees}$, i.e., rooted labelled trees $T$, where the labels $ v_1, v_2, v_3, \ldots$ on every path starting at the root of $T$ satisfy $v_1 < v_2 > v_3 < v_4 > \cdots$. First we consider various tree families of interest in combinatorics (such as unordered, ordered, $d$-ary and Motzkin trees) and study the number $T_n$ of different up-down alternating labelled trees of size $n$. We obtain for all tree families considered an implicit characterization of the exponential generating function $T(z)$ leading to asymptotic results of the coefficients $T_n$ for various tree families. Second we consider the particular family of up-down alternating labelled ordered trees and study the influence of such an alternating labelling to the average shape of the trees by analyzing the parameters $\textit{label of the root node}$, $\textit{degree of the root node}$ and $\textit{depth of a random node}$ in a random tree of size $n$. This leads to exact enumeration results and limiting distribution results. Nous étudions deux problèmes de dénombrement d'$\textit{arbres alternés haut-bas}$ : par définition, ce sont des arbres munis d'une racine et tels que, pour tout chemin partant de la racine, les valeurs $v_1,v_2,v_3,\ldots$ associées aux nœuds du chemin satisfont la chaîne d'inégalités $v_1 < v_2 > v_3 < v_4 > \cdots$. D'une part, nous considérons diverses familles d'arbres intéressantes du point de vue de l'analyse combinatoire (comme les arbres de Motzkin, les arbres non ordonnés, ordonnés et $d$-aires) et nous étudions pour chaque famille le nombre total $T_n$ d'arbres alternés haut-bas de taille $n$. Nous obtenons pour toutes les familles d'arbres considérées une caractérisation implicite de la fonction génératrice exponentielle $T(z)$. Cette caractérisation nous renseigne sur le comportement asymptotique des coefficients $T_n$ de plusieurs familles d'arbres. D'autre part, nous examinons le cas particulier de la famille des arbres ordonnés : nous étudions l'influence de l'étiquetage alterné haut-bas sur l'allure générale de ces arbres en analysant trois paramètres dans un arbre aléatoire (valeur de la racine, degré de la racine et profondeur d'un nœud aléatoire). Nous obtenons alors des résultats en terme de distribution limite, mais aussi de dénombrement exact.

scholarly journals BOUNDARY ANGULAR DERIVATIVES OF GENERALIZED SCHUR FUNCTIONS

2012 ◽  
Vol 93 (3) ◽  
pp. 203-224
Author(s):  
VLADIMIR BOLOTNIKOV ◽  
TENGYAO WANG ◽  
JOSHUA M. WEISS
Keyword(s):  
Schur Functions ◽  
Taylor Coefficients ◽  
Generalized Schur Functions ◽  
Angular Derivatives ◽  
Derivatives Of

AbstractCharacterization of generalized Schur functions in terms of their Taylor coefficients was established by Krein and Langer [‘Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume ${\Pi }_{\kappa } $ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen’, Math. Nachr. 77 (1977), 187–236]. We establish a boundary analogue of this characterization.

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