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GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable
, 1992
"... We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package. ..."
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Cited by 140 (18 self)
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We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.
Analytic Variations on QuadTrees
, 1991
"... Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently ..."
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Cited by 28 (4 self)
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Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently drawn coordinate values. The analysis leads to a class of "fullhistory" divideandconquer recurrences. These recurrences are solved using generating functions, either exactly for dimension d = 2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions relie on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques. These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.
Search costs in quadtrees and singularity perturbation asymptotics
 Discrete Comput. Geom
, 1994
"... Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a ..."
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Cited by 21 (5 self)
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Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions 1.
On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s Duke
 Math. J
, 1998
"... In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions ..."
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Cited by 17 (15 self)
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In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions
On The Borel Summability Of Divergent Solutions Of The Heat Equation
 Nagoya Math. J
, 1997
"... this paper is to improve the asymptotic results by discussing the summation of formal solutions ..."
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Cited by 16 (0 self)
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this paper is to improve the asymptotic results by discussing the summation of formal solutions
On the nonholonomic character of logarithms, powers, and the nth prime function
, 2005
"... We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction ..."
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Cited by 16 (6 self)
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We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularitybased methods and several naturally occurring sequences are proved to be nonholonomic.
Linear meromorphic differential equations: a modern point of view
 Bulletin (New Series) of the American Mathematical Society, 33(1):1 – 42
, 1996
"... Abstract. A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substant ..."
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Cited by 16 (1 self)
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Abstract. A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems. 1.