We know ((6), p. 343) that

and

where α,β,α′ and β′ are parameters such that

Consider

Recently (2) I gave certain transformations involving cognate trigonometrical series of the hypergeometric type. The arguments of the ordinary bilateral hypergeometric series involved were unity. In this paper, using the same technique applied in the above paper some general transformations have been established which further generalize the previous results.

Following the usual notation for generalized hypergeometric functions we let

(a) denotes the sequence of A parameters

that is, there are A of the a parameters and B of the b parameters. Thus ((a))m has the interpretation

with a similar interpretation for ((b))m; Δ(k; α) stands for the set of k parameters

and for the sake of brevity, the pair of parameters like α + β, α − β will be written as α ± β, the gamma product Γ(α + β) Γ(α − β) as Γ(α ± β), and so on.

The aim of this note is to obtain some formulae for generalized Rice polynomials as defined by (2)

Kampé de Fériet (l) has defined a generalized hypergeometric function of two variables as

where ∏(σp)s stands for the product (σ1)s (σ2)s … (σp)s.

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equation

Here ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equation

on the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

It is shown that a singular integral equation, arising in the theory of neutron streaming in cylindrical channels, can be solved by the Wiener-Hopf technique. The singular equation is converted into a non-singular Fredholm one, which may be solved very simply by finite difference methods and matrix inversion.

The dimer problem, which in the three-dimensional case is one of the classical unsolved problems of solid-state chemistry, can be formulated mathematically as follows. We define a brick to be a d-dimensional (d ≥ 2) rectangular parallelopiped with sides whose lengths are integers. An n-brick is a brick whose volume is n; and a dimer is a 2-brick. The problem is to determine the number of ways of dissecting an n-brick into dimers; and since this is only possible when n is even we confine attention hereafter to n-bricks with n even. Consider an n-brick with sides of length a1, a2, …, ad, where n = a1a2 … ad, and write a = (a1, a2, …, ad). Let fa denote the number of ways of dissecting this brick into ½n dimers. On the basis of physical and heuristic arguments chemists have known for many years that fa increases more or less exponentially with n; and recently a rigorous proof (1) of this fact has been given in the following form: if ai → ∞ for all i = 1, 2, …, d, then n−1 logfa tends to a finite limit, which we denote by λd. The principal outstanding problem for chemists is to determine the numerical value of λ3, or failing an exact determination to estimate λ3 or to find upper and lower bounds for it.

In a recent paper (1), the author has discussed how certain ergodic properties of a doubly infinite non-negative matrix T = {tij} are related to those of its finite (n × n) corner truncations, (n)T = {(n)tij}.

The goodness of fit of hypothetical discriminant functions has been discussed by Bartlett(1), Kshirsagar(2, 3), Radcliffe(4) and Williams (5–8). The associated problem of obtaining confidence intervals for the coefficients of a single discriminant function or canonical variable has been dealt with by Bartlett (1).

Consider a queueing system GI/G/∞ in which (i) the inter-arrival times are distributed with distribution function A(t) (A(O +) = 0) (ii) the service times have distribution function B(t) such that the expected value of the service time is β(>∞).

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given by

where a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense that

Set qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)

where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence of

to zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

The determinantal factor on p. 512 of Proc. Cambridge Philos. Soc. 61 (1965), should have the off-diagonal element in position (s, t) of the array multiplied by bst. This slip was pointed out to me by Dr Donald Knuth. When the rooted-labelled tree is not necessarily ‘chromatic,’ so that the bss's can be 1, the diagonal terms of the matrix become

The exterior metric of an infinitely long cylinder whose parameters vary periodically along its length is obtained as a Weyl type solution in terms of Bessel functions, and extended into the interior of the source. The question of inductive energy transfer between two co-axial cylinders of this nature is considered.

In a recent paper Srivastav (2) considered the solution of certain two-dimensional mixed boundary-value problems in a wedge-shaped region. The problems were formulated as dual integral equations involving Mellin transforms and were reduced to the solution of a Fredholm integral equation of the second kind. In this paper it will be shown that a closed form solution to the problems treated in (2) may be obtained by elementary means.

Consider an inviscid incompressible fluid confined within an infinite tube of radius b and assume that its motion initially consists of a uniform translation with velocity U and a uniform rotation, with angular velocity ω, about the tube axis. Suppose now that at time t = 0 a unit source of fluid situated at a point O of the axis begins to emit fluid at a uniform rate and that ultimately the flow pattern so modified becomes steady (at a fixed point as t → ∞). Under the further assumption that far upstream flow conditions are undisturbed by the source the stream function ψ of the disturbance satisfies

where z denotes distance along the axis from O, r distance from the axis and k = 2ω/U.

Consideration is given to the flow of a micropolar liquid down an inclined plane. The steady state is analysed and Yih's technique is employed in an investigation of the stability of this flow with respect to long waves. Detailed calculations are given for thin films and it is shown that the micropolar properties of the liquid play an important role in the stability criterion.

The study of electrostatic effects on the motion of fluids goes back to the work of Rayleigh (l) who discussed the effect of surface charges on the vibration of spherical drops, and Bassett (2) who studied the effect of a radial electric field on the stability of a jet. The subject has recently been taken up again by Melcher (3) and Crowley (4), and the author became interested in the subject through the monograph of the former author in which a wide variety of wave problems are described.

The paper studies the motion of an electro-conductive non-viscous incompressible fluid in a generator of rectangular section having finite electrodes and the rest of its walls being insulating. It is assumed that in the electrodes region a homogeneous magnetic field is applied, whose orientation is normal to the plane of the motion. If the induced magnetic field is neglected, the current function ψ(x, y) and the electrical potential ϕ(x, y) satisfy a system of equations with partial derivatives of the second order, with distributions. Since the system depends upon the parameter λ = RM RH a method of successive approximations is developed, which permits one to obtain the exact solution of the problem. In the last paragraph the results are extended to the case of a generator with N pairs of symmetrical electrodes.