The purpose of this note is to complement some recent results of R. O. Davies and H.G.Eggleston.

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.

The purpose of this note is to extend the result which I established recently (see (3)) on the Diophantine equation

to some further equations of a similar kind. The following theorems will be proved.

In several recent papers G. L. Lunc (7, 8, 9, 10, 11) has examined the series

where {mn} is a sequence of positive integers or zeros, and μn > 0, μn ↑ ∞. The discussion has been aimed towards extending results which are well-known for Dirichlet series to series (1) which Lunc has very reasonably called a Taylor–Dirichlet series.

Since Riemann's dissertation of 1851, Riemann surfaces have for the most part been considered as suitable domains of definition for analytic functions. Here, however, we view them as topological spaces associated with certain kinds of field extension, and consider how their topological properties are connected with algebraic properties of these field extensions. A specialization of these results gives us arithmetic properties of fields of algebraic functions of one real variable.

This paper is a contribution to the verification of conjectures of Birch and Swinnerton-Dyer about elliptic curves (1). The evidence that they produce is largely derived from curves with complex multiplication by i. In a previous paper (8), we had considered curves with complex multiplication by √ − 2. Here we shall look at the case when the ring of complex multiplications is isomorphic to the ring Z[ω], where ω3 = 1, ω ≠ 1.

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the form

where x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

Definition A. The serieswith partial sumUn (or the sequence {Un}) is said to be summable by logarithmic means to the sum U, if

The study of the Riccati equation

plays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:

Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that

(i) if β > β0 then z → − ∞ as x → + ∞;

(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);

(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.

Moreover, β0(α) is a continuous monotone increasing function of α.

In 1914 Whittaker(12) conjectured that the Heun differential equation is the simplest equation of Fuchsian type whose solution cannot be represented by a contour integral; instead the nearest approach to such a solution is to find a homogeneous integral equation satisfied by a solution of the differential equation. In this paper we reconsider Whittaker's conjecture and show that in fact solutions of Heun's equation can be represented in terms of contour integrals, similar to those of Barnes for the hypergeometric equation. The integrands of these integrals are of a rather complicated nature and cannot be said to involve known or simpler functions although they do provide expressions for the analytic continuation of Heun functions analogous to those for the hypergeometric functions.

Professor H. Davenport has raised the following question. Let f(X) be a homogeneous form in n variables X, of degree k, with coefficients in a field K. Is it possible to write f(X) in the form

where the Li(X) are linear forms over K and N depends only on K, n, and k? In particular, what can be said when K = R, the reals; K = Q, the rationals; K = C, the complex numbers ?

We shall show that when k is odd such a representation is always possible and we shall obtain an explicit bound for N. However, if k is even, no such result is possible when K = R or Q, even when we impose the obvious necessary condition that f(X) be positive semi-definite. In fact we shall exhibit a construction for obtaining forms f(X) which are positive definite but cannot be expressed in the form

r even, where φi(X) is a homogeneous form of degree s and sr = k. We shall determine precisely which definite forms can be expressed in the required form. Some natural generalizations of the problem are then considered.

The generalized hypergeometric polynomial ((7), equation (2·1)) has been defined by

where the symbol Δ(δ, −n) represents the set of δ-parameters:

and δ, n are positive integers. The polynomial is in a generalized form which yields many known polynomials on specializing the parameters.

MacRobert (4–7) and Ragab(8) have summed many infinite and finite series of E-functions by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration. Verma (9) has given two general expansions involving E-functions from which, in addition to some new results, all the expansions given by MacRobert and Ragab can be deduced. Proceeding similarly, we have studied general summations involving H-functions. The H-function is the most generalized form of the hypergeometric function. It contains a vast number of well-known analytic functions as special cases and also an important class of symmetrical Fourier kernels of a very general nature. The H-function is defined as (2)

where 0 ≤ n ≤ p, 1 ≤ m ≤ q, αj, βj are positive numbers and aj, bj may be complex numbers.

1. Recently Ta li(10) Buschman(2, 3), Erdelyi(4) and Shrivastava(8, 9) obtained solutions of integral equations involving polynomial kernels in the range of integration x to 1. Widder(12) obtained an inversion of a convolution transform with a Laguerre polynomial as kernel.

Let Σan be a given infinite series with the sequence of partial sums {Sn}. Let the sequence be defined by

where is given by

In this paper we have obtained a transformation of an infinite series of G-function. In section 3 we have shown how such transformations are useful to sum certain series of G-function and E-function (2).

Expansions in series of functions are one of the most important tools of the applied mathematician, particularly expansions in series of the classical orthogonal polynomials, e.g. Laguerre, Jacobi and Hermite polynomials. In applied problems, the uniqueness of the particular expansion is usually intrinsic to the analysis, and often implicitly assumed. Indeed, in those cases where the functions in the series are orthogonal, uniqueness can often be proved by an argument that runs as follows. Let {φn(x)} (n = 0, 1, 2, …) be a sequence of functions orthogonal with respect to the weight function ρ(x) over the interval [0, 1], and suppose that

the series being boundedly convergent for 0 ≤ x ≤ 1.

1. Recently in these proceedings we proved that, if ℜ(α) > 1 and ℜ(α) > − 1, then ((10), p. 1088)

where F2 denotes the Appell function (see, e.g., (8), p. 211),

with, as usual,

and for covergence of the double series,

and x⇌y indicates the presence of a second term that originates from the first by interchanging x and y.

In an attempt to give extensions of certain results in the theory of Mac-Robert's E-function and Meijer's (G-function, the integrals

and

are evaluated, for positive integral values of n, in terms of Agarwal's and their numerous interesting particular cases are deduced. The scope of a further generalization of (i) and (ii), with the aid of the Mellin inversion formula, is also discussed.

It is observed that the integral (ii) provides an elegant generalization of some of the recent results of Srivastava which, in turn, incorporate a well-known formula due to Slater.

1. In this note a definite integral involving the product of Meijer's G-function and a generalized hypergeometric function is evaluated with the help of finite difference operators E. As the generalized hypergeometric function and Meijer's G-function may be converted into a number of polynomials and higher transcendental functions, the integral leads to a generalization of many results. The results given by Agrawal ((l), integral 4), Sharma ((7), integral 1) in these proceedings, Bhonsle ((2), p. 188) and Saxena ((6), p. 198) follow as particular cases of our formula. In what follows v, μ, m and h are positive integers and either of v and μ may be zero. The symbol Δ(m, a) denotes a set of m parameters a/m, (a + 1 )/m, …, (a + m − 1 )/m.