If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,

If we now regard λ as a parameter and z as the independent variable then (1.1) can be written

Supposing, therefore, that a function is of the form then

Let xik (i, k=1, 2, …, n) be n2 independent variables, aik be n2 constants, and .

Let x=(xik) be the determinant formed from the n2 elements xik, and the determinantal operator formed from the elements .

Given a series Σan, we define , by the relation

where is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,

we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

The present note contains generalisations and new proofs of certain theorems in the theory of Young Tableaux and Invariant Matrices. For an account of Young Tableaux and their applications, and an introduction to the method of Clebsch-Aronhold symbols, reference should be made to Rutherford [1], and Turnbull [1], respectively. An invariant matrix T(A) of a given square matrix A is, as appears from the context in § 4 below, a matrix of polynomials in the elements of A, regarded as independent variables, such that T(AB) = T(A) T(B). Further details, and references to original sources, are given in Wallace [1].

1. Let [ast] (s, t=0, 1, … n) be a square matrix of order n+1 and determinant |ast| and suppose that by repeated “isolation” of the variables the corresponding bilinear form has been expressed as

where, for all r,

Then

Now (1) implies, and is implied by, the identities

Thus, from any known identity of the form (4), subject to the condition (2), we may at once infer, using (3). the value of the corresponding determinant |ars|.

The question of the reflection of a wave by a cylindrical mirror is of interest in a number of fields. It is a problem in which it is difficult to obtain an expression for the reflected or scattered field without recourse to physical assumptions which are sometimes somewhat dubious. An attempt was made by Sommerfeld to solve the problem of a plane wave incident upon such a perfectly conducting mirror by means of what he termed the “Non-Final Determination of Coefficients”. Unfortunately, a close examination of the problem renders it doubtful whether the method can be legitimately employed. It is possible, however, to solve the problem by expressing the scattered field in terms of the currents produced in the mirror, and finding the current generated in the mirror by an arbitrary incident field. The problem which we shall consider is the following two- dimensional one.

1. Let w(x) be a non-negative weight function for the finite interval (a, b) such that exists and is positive, and let Tr(x), r = 0, 1, 2,…be the corresponding orthonormal system of polynomials. Then if F(x) is continuous on (a, b) and has “Fourier” coefficients

Parseval's formula gives

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the series

When |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).

Suppose that un ≥ un+1 > 0, for n = 1, 2, … and that e1, e2, … are factors which make convergent.

Let where all the a's are positive, diverges, and Define En by the equation

For the case where an = 1 it has been shewn by Fuchs1 and Karamata1 that, under various conditions, En = 0(1) The object of this note is to extend some of their results.