scholarly journals A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

10.37236/1129 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Eric Fusy ◽  
Xavier Gourdon ◽  
Daniel Panario ◽  
Nicolas Pouyanne
Keyword(s):  
Unit Circle ◽  
Hybrid Method ◽  
Generating Functions ◽  
Infinite Product ◽  
Singularity Analysis ◽  
Natural Boundary ◽  
Moderate Growth ◽  
Analysis Theory ◽  
Periodic Fluctuations ◽  
Darboux's Method

A "hybrid method", dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

A General Asymptotic Scheme for the Analysis of Partition Statistics

2014 ◽  
Vol 23 (6) ◽  
pp. 1057-1086 ◽  
Author(s):  
PETER J. GRABNER ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Generating Function ◽  
Asymptotic Expansions ◽  
Generating Functions ◽  
Singularity Analysis ◽  
Higher Moments ◽  
Integer Partitions ◽  
Classical Singularity ◽  
Random Integer ◽  
Partition Statistics ◽  
New Statistics

We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

scholarly journals Markov chains with quasitoeplitz transition matrix: first zero hitting

1989 ◽  
Vol 2 (3) ◽  
pp. 205-216
Author(s):  
Alexander M. Dukhovny
Keyword(s):  
Markov Chains ◽  
Boundary Value Problems ◽  
Unit Circle ◽  
Generating Functions ◽  
Transition Matrix ◽  
Boundary Value ◽  
Riemann Boundary ◽  
Hitting Probabilities

This paper continues the investigation of Markov Chains with a quasitoeplitz transition matrix. Generating functions of first zero hitting probabilities and mean times are found by the solution of special Riemann boundary value problems on the unit circle. Duality is discussed.

INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

2015 ◽  
Vol 91 (3) ◽  
pp. 400-411 ◽  
Author(s):  
WILLIAM DUKE ◽  
HA NAM NGUYEN
Keyword(s):  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Unit Circle ◽  
Roots Of Unity ◽  
Cyclotomic Polynomials ◽  
Natural Boundary ◽  
Infinite Products

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

scholarly journals SINGULARITY ANALYSIS OF GENERATING FUNCTIONS

2011 ◽  
pp. 375-438 ◽  
Author(s):  
Philippe Flajolet ◽  
Robert Sedgewick
Keyword(s):  
Generating Functions ◽  
Singularity Analysis

scholarly journals Isomorphism and Symmetries in Random Phylogenetic Trees

2009 ◽  
Vol 46 (04) ◽  
pp. 1005-1019 ◽  
Author(s):  
Miklós Bóna ◽  
Philippe Flajolet
Keyword(s):  
Phylogenetic Tree ◽  
Gaussian Distribution ◽  
Generating Functions ◽  
Phylogenetic Trees ◽  
Singularity Analysis ◽  
Square Root ◽  
Analytic Combinatorics ◽  
Large Size ◽  
Polynomial Factor

The probability that two randomly selected phylogenetic trees of the same size are isomorphic is found to be asymptotic to a decreasing exponential modulated by a polynomial factor. The number of symmetrical nodes in a random phylogenetic tree of large size obeys a limiting Gaussian distribution, in the sense of both central and local limits. The probability that two random phylogenetic trees have the same number of symmetries asymptotically obeys an inverse square-root law. Precise estimates for these problems are obtained by methods of analytic combinatorics, involving bivariate generating functions, singularity analysis, and quasi-powers approximations.

scholarly journals The Kostant partition functions for twisted Kac-Moody algebras

2000 ◽  
Vol 24 (1) ◽  
pp. 11-27
Author(s):  
Ranabir Chakrabarti ◽  
Thalanayar S. Santhanam
Keyword(s):  
Generating Functions ◽  
Infinite Product ◽  
Triple Product ◽  
Partition Functions ◽  
Recursion Relations

Employing the method of generating functions and making use of some infinite product identities like Euler, Jacobi's triple product and pentagon identities we derive recursion relations for Kostant's partition functions for the twisted Kac-Moody algebras.

scholarly journals Singularity Analysis Via the Iterated Kernel Method

2014 ◽  
Vol 23 (5) ◽  
pp. 861-888 ◽  
Author(s):  
STEPHEN MELCZER ◽  
MARNI MISHNA
Keyword(s):  
Generating Functions ◽  
Efficient Algorithm ◽  
Kernel Method ◽  
Singularity Analysis ◽  
Lattice Path ◽  
Exact Enumeration ◽  
Quarter Plane ◽  
Path Models

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

An integral Representation of a 10ϕ9 and Continuous Bi-Orthogonal 10ϕ9 Rational Functions

1986 ◽  
Vol 38 (3) ◽  
pp. 605-618 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Integral Representation ◽  
Unit Circle ◽  
Infinite Product ◽  
Rational Functions ◽  
Positive Direction ◽  
Beta Integral ◽  
Image Position

1. Introduction. One of the most remarkable q-extensions of the classical beta integral was recently introduced by Askey and Wilson [1](1.1)where |q| < 1 and the pairwise products of {a, b, c, d} as a multiset do not belong to the set {qj, j = 0, – 1, – 2, …}. The contour C is the unit circle described in the positive direction, but with suitable deformations to separate the sequences of poles converging to zero from the sequences of poles diverging to infinity. The symbol (A; q)∞ is an infinite product defined by(1.2)whenever it converges.

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