A number of formulae are known which exhibit the asymptotic behaviour as t→∞ of the solutions of

The aim of thisnote is to unify a group of such formulae, relating to the case in which F(t) iS on the whole positive, and suitably continuous though not necessarily analytic.

In this paper we evaluate a few infinite integrals involving products of Legendre functions. The results obtained herein are quite general and include, as particular cases, some known results.

The E-functions were defined by MacRobert [3] in 1937; they are denoted by E (p; αr: q; ps. z).

In § 3 of this paper, I prove a new expansion for E(p; αr: q; ps: z) which is similar to an expansion due to MacRobert [2], viz.,

where

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.

The Mathieu functions of integral order [1] are the solutions with period π or 2π of the equation

The eigenvalues associated with the functions ceN and seN, where N is a positive integer, denoted by aN and bN respectively, reduce to

aN = bN = N2

when q is zero. The quantities aN and bN can be expanded in powers of q, but the explicit construction of high order coefficients is very tedious. In some applications the quantity of most interest is aN – bN, which may be called the “width of the unstable zone“. It is the object of this note to derive a general formula for the leading term in the expansion of this quantity, namely

Suppose first that N is an odd integer. Then there is an expansion

where

These functions π satisfy

and

On Substituting (3) in (1), one obtains the algebraic equation

where

Explicitly,

{11} = q

{lm} = 0 otherwise.

Let A be a closed subset of a topological space X and f a continuous mapping of A into X with the following two properties:

1.1. f {Fr (A)} & f {Int (A)} are disjoint.

1.2. The mapping f* = f | Fr (A) is 1–1.

It is proved in [5], that if X is the euclidean n–sphere Sn = {x; x e Rn+1 and ― x ― = 1}, then

1.3. f {Fr(A)} = Fr {f(A)}.

[ Hence f{Int (A)} = Int {f(A)}].

Darling [3] in 1932 and Bailey [2] in 1933 gave certain theorems on products of hypergeometric series. Again in 1948 Sears [4] used the relation which expresses the series in terms of M other series of the same type to derive transformations between products of both basic and ordinary hypergeometric series. In this paper I give certain general theorems on products of bilateral hypergeometric series together with some of their interesting special cases.

In his fundamental paper, “On the structure of semigroups” [6], J. A. Green has examined certain important minimal conditions which may be satisfied bya semigroup S.We say that S satisfies the minimal condition on principal left ideals if every set of principal left ideals of S contains a minimal member with respect to inclusion:this condition is denoted by ℳ1. The corresponding conditions on principal rightideals and principal two-sided ideals are denoted by ℳr and ℳ1 respectively. The purpose of the present paper is to give some further results concerning these three conditions.Extensive use is made of the work of A. H. Clifford ([3] and [4]) onminimal ideals.