Our main result is that the fourth power of the determinant of an arbitrary m by m persymmetric matrix is identically equal to the determinant of a certain 2m by 2m skew-symmetric matrix. Having proved this, we derive several other identities involving powers of persymmetric determinants.

It is well known that every representation of the group Un of unitary matrices of order n × n is equivalent to a unitary representation (see e.g. Little-wood (6), ch. XI). Our object in the present paper is to discuss some properties of those representations, and to construct a specific unitary representation.

A graph G is a pair (V{G), E(G)); V(G) is the set of vertices of G, and E(G) the set of edges of G; E(G) is a subset of the Cartesian product V(G) × V(G) (all pairs (υi, υj) where υi∈ V(G), υj ∈ V(G)). If υ is a vertex of G we shall write υ ∈ G. The graph-theoretic terminology used in this paper is that of Berge (1). We shall consider only graphs which are finite, undirected, and without loops.

The subject of squared rectangles and squared squares was developed by Brooks, Smith, Stone and Tutte(1) using techniques based on the theory of electrical networks. In this note we shall treat the (apparently) more general topic of rectangulations using purely algebraic techniques. The starting point is the observation that the incidence matrices of the pair of dual networks associated with a squared rectangle in (l) can be more naturally derived as incidence matrices of the squaring (or rectangulation) itself. Further, the topological information contained in the networks is a simple consequence of Euler's formula applied directly to the rectangulation.

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.

If (X, g) is a paranormed space, with paranorm g (see (2)), then we denote by X* the continuous dual of X, i.e. the set of all continuous linear functionals on X. If E is a set of complex sequences x = (xk) then E† will denote the generalized Köthe–Toeplitz dual of E

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.

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 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).

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.

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.

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).

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.

We have defined the generalized hypergeometric polynomial ((6), eqn. (2·1), p. 79) by means of

where δ and n are positive integers and the symbol Δ(δ, − n) represents the set of δ-parameters

The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters and therefore the results are of general character.

1·1. Definitions and notations. Let ∑an be a given infinite series, and let sn be its nth partial sum. We denote by and the nth Cesàro means of order α(α > − 1) of the sequences {sn} and {n. an} respectively.

Let ∑an be a given infinite series and {sn} the sequence of its partial sums. Let {pn} be a sequence of constants, real or complex, and let us write

In this paper we are investigating boundedness and certain stability properties of differential systems in spaces, utilizing the generalization of Bellman's Lemma which was formulated by one of the authors (8).

With a view to applications to electrical engineering we have recently calculated the primitive polynomials of degree 2 and 3 modulo various primes. In the course of the work we also obtained all the irreducible polynomials of these kinds and, as an extension of the tables of Church(1), Marsh(2), Peterson(3) and Lloyd(4) we now publish them. Since our primary interest was the exponent to which each polynomial belonged (see Albert(5), p. 130) we group them according to exponent.

This paper is concerned with a generalization to n dimensions of a method for obtaining an approximation to a root of an algebraic or transcendental equation. We prove a convergence theorem for the method, and show how the method becomes of considerable practical utility in the case when the Jacobian matrix of the functions constituting the system is strictly diagonally dominant in a neighbourhood.