We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ s ≤ t} of a growth-collapse process (Xt )t≥0, defined as a [0,∞)-valued Markov process that grows linearly between random ‘collapse’ times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in term...

For a regular chain R, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of R, that is, the set W (R) \W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We focus on the case where this saturated ideal has dimension one and we discuss extensions of this work in hi...

Using a simple transfer matrix approach we have derived long series expansions for the perimeter generating functions of both three-choice polygons and punctured staircase polygons. In both cases we find that all the known terms in the generating function can be reproduced from a linear Fuchsian differential equation of order 8. We report on an analysis of the properties of the differential equ...

Beginning in 1893, L. J. Rogers produced a collection of papers in which he considered series expansions of infinite products. Over the years, his identities have been given a variety of partition theoretic interpretations and proofs. These existing combinatorial techniques, however, do not highlight the similarities and the subtle differences seen in so many of these remarkable identities. It ...

We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems. The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced. Its construction was extended to the nonreduced case by the author. It is a meromorphic We...

For a regular chain R in dimension one, we propose an algorithm which computes the (non-trivial) limit points of the quasi-component of R, that is, the set W (R) \W (R). Our procedure relies on Puiseux series expansions and does not require to compute a system of generators of the saturated ideal of R. We provide experimental results illustrating the benefits of our algorithms.

The finite-lattice method of series expansions has been combined with a new extrapolation procedure to extend the low-temperature series for the specific heat, spontaneous magnetisation and susceptibility of the spin-1 Ising model on the square lattice. The extended series were derived by directly calculating the series to order 99 (in the variable u = exp[−J/k B T ]) and using the new extrapol...

After reviewing some aspects of Cohn’s theory of the free field, we give another proof of a theorem of Lewin: the subfield of rational elements in the field of Malcev-Neumann series on the free group is isomorphic to the free field.

Let Jμ denote the Bessel function of order μ. The functions x−α−1Jα+2n+1(x), n = 0, 1, 2, . . . , form an orthogonal system in the space L2((0,∞), x2α+1dx) when α > −1. In this paper we prove that the Fourier series associated to this system is of restricted weak type for the endpoints of the interval of mean convergence, while it is not of weak type if α ≥ 0.

This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis for factorizations other than the well-known binary and ternary basis. We show that is possible to reduce the overall complexity for the usual binary decompos...