Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , x, . . . } to get a sequence {Pn}∞n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of p...

From power series expansions of functions on curves over finite fields, one can obtain sequences with perfect or almost perfect linear complexity profile. It has been suggested by various authors to use such sequences as key streams for stream ciphers. In this work, we show how long parts of such sequences can be computed efficiently from short ones. Such sequences should therefore considered t...

Dzhaparidze and van Zanten (2004, 2004b) proved an explicit series expansion of the multi-parameter fractional Brownian sheet. We extend their results to a certain class of isotropic Gaussian random fields with homogeneous increments, In particular, this class contains the multi-parameter fractional Brownian motion. Let ξ(x) be a centred mean-square continuous Gaussian random field with homogen...

Droplets on insulating material suffer a nonvanishing total ponderomotive force because of the inhomogeneity of the surrounding electric field. A series expansion of this total force is proven in a two-dimensional setting by determining the line charge density at the boundary of the test body via a Fredholm integral equation, which is solved by Fourier techniques. The influence of electric char...

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.