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DFiniteness: Algorithms and Applications
 Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation
, 2005
"... Differentially finite series are solutions of linear differential equations with polynomial coefficients. Precursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their ..."
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Differentially finite series are solutions of linear differential equations with polynomial coefficients. Precursive sequences are solutions of linear recurrences with polynomial coefficients. Corresponding notions are obtained by replacing classical differentiation or difference operators by their qanalogues. All these objects share numerous properties that are described in the framework of “Dfiniteness”. Our aim in this area is to enable computer algebra systems to deal in an algorithmic way with a large number of special functions and sequences. Indeed, it can be estimated that approximately 60 % of the functions described in Abramowitz & Stegun’s handbook [1] fall into this category, as well as 25% of the sequences in Sloane’s encyclopedia [20, 21]. In a way, Dfinite sequences or series are noncommutative analogues of algebraic numbers: the role of the minimal polynomial is played by a linear operator. Ore [14] described a noncommutative version of Euclidean division and extended Euclid algorithm for these linear operators (known as Ore polynomials). In the same way as in the commutative case, these algorithms make several closure properties effective (see [22]). It follows that identities between these functions or sequences can be proved or computed automatically. Part of the success of the gfun package [17] comes from an implementation of these operations. Another part comes from the possibility of discovering such identities empirically, with PadéHermite approximants on power series [2] taking the place of the LLL algorithm on floatingpoint numbers. The discovery that a series is Dfinite is also important from the complexity point of view: several operations can be performed on Dfinite series at a lower cost than on arbitrary power series. This includes multiplication, but also evaluation at rational points by binary splitting [4]. A typical application is the numerical evaluation of π in computer algebra systems; we give another one in these proceedings [3]. Also, the local behaviour of solutions of linear differential equations in the neighbourhood of their singularities is well Copyright is held by the author/owner.
Automatic enumeration of regular objects
 J. Integer Sequences
"... Abstract. We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the ..."
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Abstract. We describe a framework for systematic enumeration of families combinatorial structures which possess a certain regularity. More precisely, we describe how to obtain the differential equations satisfied by their generating series. These differential equations are then used to determine the initial counting sequence and for asymptotic analysis. The key tool is the scalar product for symmetric functions and that this operation preserves Dfiniteness.
ANALYTIC ASPECTS OF THE SHUFFLE PRODUCT
, 2008
"... There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects o ..."
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There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of Dfinite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models Dfinite generating functions.
HAYMAN ADMISSIBLE FUNCTIONS IN SEVERAL VARIABLES
"... Abstract. An alternative generalisation of Hayman’s admissible functions ([17]) to functions in several variables is developed and a multivariate asymptotic expansion for the coefficients is proved. In contrast to existing generalisations of Hayman admissibility ([7]), most of the closure properties ..."
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Abstract. An alternative generalisation of Hayman’s admissible functions ([17]) to functions in several variables is developed and a multivariate asymptotic expansion for the coefficients is proved. In contrast to existing generalisations of Hayman admissibility ([7]), most of the closure properties which are satisfied by Hayman’s admissible functions can be shown to hold for this class of functions as well. 1.
FOR THE DOCTORATE IN MATHEMATICS BY
"... NOVEMBER 2003This thesis is dedicated to the memory of a wonderful woman, my mother, Vicki Munn. Acknowledgements I offer the following people and organizations heartfelt appreciation for their contributions to this work and their support during the period of my thesis. The two research teams which ..."
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NOVEMBER 2003This thesis is dedicated to the memory of a wonderful woman, my mother, Vicki Munn. Acknowledgements I offer the following people and organizations heartfelt appreciation for their contributions to this work and their support during the period of my thesis. The two research teams which I had the pleasure of being part: LaCIM (Université du Québec à Montréal) and Projet Algorithmes (Inria, Rocquencourt, France). Both are thriving incubators of combinatorics with warm, human aspects, and regular coffee. I thank also the students and postdocs of both groups, such as Marianne Durand, Cédric Lamathe, and Ludovic Meunier, for their humour and camaraderie; A special separate mention must be made for les assistantes extraordinaires Lise
KRONECKER PRODUCT IDENTITIES FROM DFINITE SYMMETRIC FUNCTIONS
, 2006
"... Abstract. Using an algorithm for computing the symmetric function Kronecker product of Dfinite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for tracelike values of the Kronecker product. ..."
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Abstract. Using an algorithm for computing the symmetric function Kronecker product of Dfinite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for tracelike values of the Kronecker product.
www.stacsconf.org ANALYTIC ASPECTS OF THE SHUFFLE PRODUCT
, 2008
"... Abstract. There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key ..."
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Abstract. There exist very lucid explanations of the combinatorial origins of rational and algebraic functions, in particular with respect to regular and context free languages. In the search to understand how to extend these natural correspondences, we find that the shuffle product models many key aspects of Dfinite generating functions, a class which contains algebraic. We consider several different takes on the shuffle product, shuffle closure, and shuffle grammars, and give explicit generating function consequences. In the process, we define a grammar class that models Dfinite generating functions.