Frits Beukers

Monodromy for the hypergeometric function nFn-1.

La fonction hypergeometrique. Le groupe hypergeometrique. La forme hermitienne invariante. Le cas imprimitif. Theorie de Galois differentielle. Fonctions hypergeometriques algebriques

Another congruence for the Apéry numbers

Abstract In 1979 R. Apery introduced the numbers a n = Σ 0 n ( k n ) 2 ( k n + k ) 2 in his irrationality proof for ζ(3). We prove some congruences for these numbers, which extend congruences previously published in J. Number Theory (Vol. 12, 14, 16, and 21).

Gauss' hypergeometric function

We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hypergeometric equation.

Some Congruences for the Apery Numbers

Abstract In 1979 R. Apery introduced the numbers an = Σ0n(kn)2(kn + k) and un = Σ0n(kn)2(kn + k)2 in his irrationality proof for ζ(2) and ζ(3). We prove some congruences for these numbers which generalize congruences previously published in this journal.

On a sequence of polynomials

Abstract In connection with studies of hierarchies of solutions to the Korteweg-de Vries equation, J. Sanders asked if gcd( p k , p l ) is trivial for all k , l , where p k = ( x + 1) k − x k − 1. In this paper we propose a positive reply to this question.

An alternative proof of the Lindemann-Weierstrass theorem

In December 1987 J. P. Bezivin and Ph. Robba found a new proof of the Lindemann-Weierstrass theorem as a by-product of their criterion of rationality for solutions of differential equations. Let us recall the Lindemann-Weierstrass theorem, to which we shall refer as LW from now on.

Unitary monodromy of Lamé differential operators

The classical second order Lamé equation contains a so-called accessory parameter B. In this paper we study for which values of B the Lamé equation has a monodromy group which is conjugate to a subgroup of SL(2,ℝ) (unitary monodromy with indefinite hermitian form). We reformulate the problem as a spectral problem and give an asymptotic expansion for the spectrum.

Duality relations for hypergeometric series

We explicitly give the relations between the hypergeometric solutions of the general hypergeometric equation and their duals, as well as similar relations for q-hypergeometric equations. They form a family of very general identities for hypergeometric series. Although they were foreseen already by N. M. Bailey in the 1930s on analytic grounds, we give a purely algebraic treatment based on general principles in general differential and difference modules.

Power Values of Divisor Sums

Abstract We consider positive integers whose sum of divisors is a perfect power. This problem had already caught the interest of mathematicians from the 17th century like Fermat, Wallis and Frenicle. In this article we study this problem and some variations.We also give an example of a cube, larger than one, whose sum of divisors is again a cube.

Factorisation of Polynomials

In many rings, commutative or not commutative, the elements can be written as a product of irreducible elements (not necessarily unique). In algorithms in computer algebra it is often essential that this should be realized in an efficient way. The most important examples in this respect are ℤ, Fq[X], ℤ[X] and, as a more recent example of computational interest, the ring ℚ(X)[d/dX] [6]. The latter ring is not commutative and factorisation into irreducibles is not unique.