The object of this paper is to evaluate some integrals involving the product of Gauss's hypergeometric function and Meijer's G-function by expressing the G-function as a Mellin–Barnes type integral and interchanging the order of integrations. The integrals are important because on specializing the parameters they lead to many results for MacRobert's E-function, Bessel, Legendre, Whittaker functions and other related functions. In what follows δ is a positive integer and Δ(δ, α) represents the set of parameters

This paper is a sequel to one with a similar title to appear in the Proceedings of the Edinburgh Mathematical Society. Explicit solutions are found for two more integral equations of similar form; and also conditions necessary and sufficient for existence, and sufficient for uniqueness, of solutions. These theorems are preceded by several preparatory theorems on fractional integrals with origin ∞, including integrals of purely imaginary order.

1. The object of the present paper is to obtain some finite integrals involving the Appell's function F4 and the H-function of Fox by utilizing the results recently given by Saxena, Sharma and Tranter respectively. The first three results proved in this paper are the extension of the results recently established by Saxena and Sharma in these proceedings. MacRobert's result follows as a very special case of one of our results. A few very interesting special cases of the main results have also been given.

In this note the author has derived three Fourier series expansions for H-function which is more general than Meijer's G-function. Some integrals and a recurrence relation for H-function have been derived as corollaries.

1. In the course of a systematic analysis of certain problems in quantum mechanics it has been observed that their exact solutions can be expressed in terms of the Appell function F2 defined by means of (see e.g. (8), p. 211)

where, as usual,

and for convergence of the double series,

An addition theorem, expressing the spherical harmonics in terms of the spherical harmonics and , is derived.

1. It was known to Euler that p(n), the number of unrestricted partitions of n into non-increasing integral parts, is generated by

with the usual convention that p(0) = 1.

A generalization of the Neville–Aitken method is described which allows the construction of interpolating functions, other than polynomials, by means of simple recurrence relations. In particular, simple constructions are given for rational functions and trigonometric series which interpolate prescribed function values at non-equispaced positions of the independent variable.

Restrictions imposed by requiring the interpolating functions to be invariant under linear transformations of the coordinates are discussed, and application of the technique to the problem of inverse interpolation is also considered.

For a given θ(0 < θ < 1) and r = 0, 1, 2, 3,… there are two related problems on the fractional parts {rθ}, namely:

the gap problem: for any φ (0 < φ < 1) to determine the gaps between the successive r for which

the step problem: for the set

rearranged in ascending order, to determine the steps into which the interval [0,1] is thereby partitioned. There are subsidiary problems of the numbers of gaps or steps of the various lengths, and also of the permutation of 1,…, N when (2) is arranged in order.

We recall that a square matrix (finite or infinite) with non-negative elements and with each row-sum and column-sum equal to 1 is called doubly-stochastic (d.s.). If each row-sum and column-sum of a non-negative square matrix is less than or equal to 1, the matrix is called doubly-substochastic (d.s.s.).

The part of combinatorial analysis which deals with systems of representatives draws on a variety of different sources. Thus the theory of graphs, the theory of flows in networks, linear programming, and integral linear programming all provide relevant methods and arguments. Nevertheless, it is in all likelihood possible to give a unified set-theoretic account of the subject, based in the main on Hall's criterion for the existence of transversals and on a certain generalization of the Schröder–Bernstein theorem. This approach has the added advantage of greater flexibility since it enables us sometimes to cope with transfinite situations where other methods may be only applicable to finite cases.

The question has arisen in the theory of queues whether the product of two independent exponentially distributed random variables is infinitely divisible. In this note it is proved that the product of any non-negative random variable with an independent exponentially distributed variable is infinitely divisible. Such random variables are members of a wider class of infinitely divisible random variables, which will be exhibited first.

An approximate formula is found for the field produced on the boundary of an obstacle in an inhomogeneous medium by a source which is close to the boundary. Both harmonic and aperiodic waves are discussed.

Energy of the exterior and the interior fields of a particle, charged or otherwise, embedded in an empty space is calculated, using Møller's tetrad formulation of the energy-momentum complex in general relativity. It is found that the exterior field contains positive gravitational energy, and in addition, the electromagnetic energy if the particle is a charged one and that the sum total of the energies of the interior and the exterior fields is equivalent to the Newtonian mass of the particle.

The solutions are derived for the shallow water standing waves of small amplitude that can form in channels or lakes of varying breadth with a concave parabolic bottom. In addition explicit solutions are given for standing waves in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls.

In this paper we obtain a representation of elastic wave fields (i.e. solutions of the equations of motion of classical elastokinetics) in terms of three functions satisfying scalar wave equations. Although the form of the representation implies no restriction upon the choice of coordinate system or upon the shape of the elastic body, it is found that the result can only be applied advantageously to initial-boundary-value problems having spherical polar coordinates as a natural frame of reference. The representation is shown to be complete in the sense that every (sufficiently smooth) elastic wave field in a homogeneous, isotropic body bounded by two concentric spheres can be expressed in the given form.

Under conditions of axial symmetry the representation generates wave fields which can be decomposed into poloidal and toroidal constituents, the former arising from two scalar wave functions and comprising both dilatational and rotational waves, and the latter being associated with a single scalar wave function and a state of pure shear of the elastic solid. Finally, the representation is used to obtain a formal solution describing the elastic pulse generated in an infinite body by the application of time-dependent tractions to the surface of a spherical cavity.

A detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.

In this paper, which is a continuation of (1), we study steady (i.e. time-harmonic) and transient rectilinear oscillations of small amplitude of a rigid sphere embedded in an infinite elastic solid. Two types of transient motions are considered, forced oscillations in which the sphere is subject to a prescribed time-dependent force, and free oscillations in which the sphere is set in motion by an impulsive force. For each mode of vibration of the sphere the character of the solution is determined by two parameters, the density contrast between the sphere and its surroundings and a parameter related to the Poisson's ratio of the elastic solid. Numerical results referring to transient rectilinear oscillations are presented in graphical form.

This paper deals with wave propagation in inhomogeneous compressible electron plasmas. The waves are described by Maxwell's equations coupled to the linearized single-fluid equations of hydrodynamics. Coupled wave equations are derived which describe the propagation of coupled electromagnetic and electron acoustic waves. Results are obtained for generally inhomogeneous plasmas and are specialized to planar and cylindrically stratified media. Particular attention is given to the planar case and several approximate techniques for treating the equations are discussed. The fields in a region of coupling are investigated.