Ramanujan's Q-function and the so called "tree function" T (z) defined implicitly by the equation T (z) = ze T (z) found applications in hashing, the birthday paradox problem, random mappings, caching, memory conflicts, and so forth. Recently, several novel applications of these functions to information theory problems such as linear coding and universal portfolios were brought to light. In this paper, we study them in the context of another information theory problem, namely: universal coding which was recently investigated by Shtarkov et al. [Prob. Inf. Trans., 31, 1995]. We provide asymptotic expansions of certain recurrences studied there which describe the optimal redundancy of universal codes. Our methodology falls under the so called analytical information theory that was recently applied successfully to a variety of information theory problems. Key Words: Source coding, multi-alphabet universal coding, redundancy, minimum description length, analytical information theory, si...

This paper introduces the framework of decomposable combinatorial structures and their traversal algorithms. A combinatorial type is decomposable if it admits a specification in terms of unions, products, sequences, sets, and cycles, either in the labelled or in the unlabelled context. Many properties of decomposable structures are decidable. Generating function equations, counting sequences, and random generation algorithms can be compiled from specifications. Asymptotic properties can be determined automatically for a reasonably large subclass. Maple libraries that implement such decision procedures are briefly surveyed (LUO, combstruct, equivalent). In addition, libraries for manipulating holonomic sequences and functions are presented (gfun, Mgfun).

We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given exp-log function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddle-point method is then treated in detail.

An ordered set-partition (or preferential arrangement) of n labeled elements represents a single "hierarchy"

We describe a simple package for computing a fundamental system of certain formal series solutions, up to a prescribed order, of a given P-finite recurrence equation. These solutions can be viewed as describing the asymptotic behavior of sequences satisfying the recurrence.

We consider an extension of Hayman's notion of admissibility to bivariate generating functions f(z; u) that have the property that the coecients a nk satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a big class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is presented, too. We apply this concept to various combinatorial examples. 1.

The enumeration of lattice paths in wedges poses unique mathematical challenges. These models are not translationally invariant, and the absence of this symmetry complicates both the derivation of a functional recurrence for the generating function, and solving for it. In this paper we consider a model of partially directed walks from the origin in the square lattice confined to both a symmetric wedge defined by Y = ±pX, and an asymmetric wedge defined by the lines Y = pX and Y = 0, where p> 0 is an integer. We prove that the growth constant for all these models is equal to 1+ √ 2, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when p = 1. From these we find asymptotic formulas for the number of partially directed paths of length n in a wedge when p = 1. The functional recurrences are solved by a variation of the kernel method, which we call the “iterated kernel method”. This method appears to be similar to the obstinate kernel method used by Bousquet-Mélou (see, for example, references [5, 6]). This method requires us to consider iterated compositions of the roots of the kernel. These compositions turn out to be surprisingly tractable, and we are able to find simple explicit expressions for them. However, in spite of this, the generating functions turn out to be similar in form to Jacobi θ-functions, and have natural boundaries on the unit circle.

We describe the current state of a Maple library, gdev, designed to perform asymptotic expansions for a large class of expressions. Many examples are provided, along with a short sketch of the underlying principles. At the time when this report is written, a striking feature of these examples is that none of them can be computed directly with any of today's most widespread symbolic computation systems (Macsyma 1 , Mathematica 2 , Maple 3 or Scratchpad II 4 ). This document is part of a technical report [4] available from INRIA. Introduction Current symbolic computation systems generally lack facilities for manipulating asymptotic expansion computations of a form more complex than the first terms of a Taylor series or a Puiseux expansion (involving fractional powers). We introduce a set of programs whose aim is to contribute to fill this gap. The emphasis here is on examples displaying the variety of difficulties a general purpose program must be prepared to encounter. With eac...