1. The concept of a unique factorization domain (UFD) has been defined, for commutative (e.g. (4) page 21) and non-commutative (1) integral domains. We take the theory a stage further here by defining a unique factorization ring (UFR), where throughout, a ring is understood to mean a commutative ring with identity, possibly containing proper zero-divisors.

It is shown that a necessary and sufficient condition that two regular polynomial matrices T, U have relatively prime determinants is that the equation TX + YU = E, where E is a constant matrix, has a unique solution with the degrees of X, Y less than the degrees of U, T respectively. This is a generalization of a well-known theorem for scalar polynomials. An alternative form for the condition in terms of the non-vanishing of a determinant, corresponding to the resultant of two scalar polynomials, is also obtained together with an equivalent determinant of lower order.

Let the generalized hypergeometric function of one variable be denoted by

where (a)m is the Pochhammer symbol ((1, 3)).

We shall point out a connexion between convolution (as defined in (5); see also (7)) and the Arens multiplication on the second dual of a Banach algebra (1, 2). This enables us to demonstrate in an easy way the existence of a commutative Banach algebra whose second dual is not commutative. Examples can also be found in (3, 9) and elsewhere.

The idea of constructing a space of functions taking values in a locally convex space E from a linear space of scalar valued functions is well known. We can, for example, define a space consisting of all E-valued functions φ(t) such that for all elements e′ of the dual E′ of E. Besides this construction there are others which arise in special cases. This idea has been used to obtain integrals of vector-valued functions (compare (2), Chapter III, § 4). Schwartz has also used it in his paper on differentiable vector-valued functions (9) whose main result is the famous kernel theorem, as well as in introducing vector-valued distributions. It is natural to expect that the space of vector-valued functions obtained will inherit some properties of the function space and the vector space E. Therefore one usually starts from some function space which has interesting properties.

Let X be an F-space, not necessarily locally convex, with a quasi-norm ‖x‖ (see (7), p. 31). The series

of elements of X is commutatively convergent if and only if converges for every permutation {Yn}nεω of {Xn}nεω.

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

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.

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.

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.

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.

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.

The object of this paper is to establish an integral involving Fox's H-function and employ it to obtain an expansion formula for the H-function involving Bessel functions.

1. It has been observed that certain problems in quantum mechanics have their exact solutions which can be expressed in terms of Appell's function F2 as defined by (e.g. (7), p. 211)

Having regard to this fact, Srivastava (5) proved a summation formula

where R(λ) > 1, R(α) > − 1 and x⇌y indicates the presence on the right of a second term that originates from the first by interchanging x and y.

1. The object of this paper is to prove some formulae of Jacobi polynomials including a generating function. The results (2·l)–(2·4), (2·6)–(2·9), (3·l)–(3·4), and (4·1) are believed to be new.

In this paper we have evaluated an integral involving Fox's H-function and Jacobi polynomial and employed its particular cases to establish two expansion formulae for the H-function involving Jacobi polynomials.

1. The object of this paper is to evaluate two integrals involving Fox's H-function and employ them to establish two Fourier series for the H-function. Some Fourier series for Meijer's G-function and MacRobert's E-function are obtained as particular cases. Some results recently given by MacRobert(6, 7), Jain (4) and Kesarwani(5) are shown as particular cases.

1. In this paper we have evaluated an integral involving Fox's H-functions and Whittaker functions. One particular case of the integral has been employed to establish an expansion formula for the H-function involving Laguerre polynomials.

In this paper the integral involving the Fox H-function and the Laguerre polynomial has been evaluated and the expansion formula for the H-function has been established with the application of this integral. Many particular results have also been given.