Explicit forms are given for the Wigner for values of J from J = 0 to J = 7/2. The tedious labour involved in evaluating available formulas is reduced by use of a computer.

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.

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

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. 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.

By using a Laplace transformation, a general solution is obtained to the problem of the oscillations and velocity field of a viscous gravitating sphere. Lamb's oscillating solution and Darwin's exponentially decaying solution are derived as asymptotic expressions and their connexion demonstrated. Closed loops of stream lines are a remarkable feature of the flow, and the conditions for their existence are discussed. Asymptotic solutions are also obtained for the oscillations of a Maxwell sphere, and their relation to those of an elastic sphere investigated.

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.

The validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.

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.

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).

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.

In this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group . These fields may be expanded as series of functions , where , m is fixed and the matrices Tl(g) form a 21 + 1 dimensional irreducible representation of .

Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with given l, n are transformed into others with the same values of l, n. That this must be so follows from Schur's Lemma and the fact that for each m and l the functions form a basis for an invariant subspace of functions on of dimension 2l + 1 in which an irreducible representation of acts. Explicit formulae for the results of such operations are presented.

The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).

This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.

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.

The first-order Chapman-Enskog (CE) approximation has been used to linearize the Boltzmann-Landau (BL) equation primarily in the binary collision approximation and a linear integral equation with a non-symmetric kernel is obtained. The solubility conditions are discussed on the basis of conservation theorems. The formal solutions and the transport coefficients have been obtained in a subsequent paper.

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.

The linearized Boltzmann-Landau transport equation given in a previous paper with a non-symmetric Kernel has been formally solved and the solution has been used to calculate the coefficients of shear viscosity (η) and thermal conductivity (λ) up to first-order density corrections.

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.

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.