Let R be a ring and G a group. The group ring RG consists of all functions f: G → R with finite support. Addition is pointwise and multiplication is defined for f, h ∊ RG and g ∊ G, by

The support group of f is defined to be the subgroup of G generated by the support of f. The element f is idempotent if ff = f

We prove the following result.

In his classic memoir on the moment problem that bears his name, Stieltjes [2] exhibited

1

as an example of an indeterminate (Stieltjes) moment sequence.

Stieltjes also obtained the corresponding S-fraction and thus implicitly obtained the three-term recurrence formula satisfied by the corresponding orthogonal polynomials.

For any two integers n and k we take, as usual,

Using the above symbol we define a new function F(q, 2n) by a sum of two finite sums given by

1.1

for i ≠ q, and whenever q is an odd integer.

Un ensemble de Wiener-Hopf est une partie S de la droite réelle R tel que

1. (i) S+ et S- sont non-vides

2. (ii) S+ + S- ⊂ S

où S - ∪ S 0 ∪ S + est la décomposition de S en parties négative, nulle, et positive.

On remarque que ces ensembles présentent une parenté avec les semi-groupes de R. Ainsi, dans [1], est-il démontré que si S n'est pas concentré sur un multiple des entiers, S est dense dans un intervalle.

In [2] it is shown that an n × n partial latin square with n — 1 cells occupied on the main diagonal can be completed to a latin square. We can use the technique in [2] to prove the following result.

Let P be an n × n partial latin square based on 0,1, 2, …, n — 1 satisfying the above conditions, and let (x 0, y 0), (x 1, y 1), …, (x n-2, y n-2) De the occupied cells where y 0, y 1, … y n-2 are distinct.

The following theorem is due to Walsh [2]. For another proof see [1].

Theorem A. All the zeros of the polynomial p(z) = a0 + a1z + … an-1zn-1+zn lie on the disk.

where M = Σj = 2 n |a n-j |1/j