Let T be a bounded symmetric operator in a Hilbert space H. According to the spectral theorem, T can be expressed as an integral in terms of its spectral family {Eλ}, each Eλ being a certain projection which is known to be the strong limit of some sequence of polynomials in T. It is a natural question to ask for an explicit sequence of polynomials in T that converges strongly to Eλ. So far as the author knows, no complete solution of this problem has been given even when H has finite dimension, i.e. when T is a finite symmetric matrix. Since a knowledge of the spectral family {Eλ} of a finite symmetric matrix carries with it a knowledge of the eigenvalues and eigenvectors, a solution of the problem may have some practical value.

It is proposed to establish the two following integrals.

where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameters

where n is a positive integer and x is real and positive.

Let A be an associative ring. Given a ∊ A, an element b ∊ A is called a left identity for a if

Given a subset S of A, an element b ∊ A is, called a left identity for S if (1) is satisfied for all a ∊ S. An element of A need not have a left identity; for example, if A is nilpotent then no non-zero element of A has a left identity. If a does have a left identity, the latter need not be unique; if every element of a subset S of A has a left identity, then it is not necessarily true that S has a left identity.

By the group G(2 cos π/q) we mean the group of linear fractional transformations of the complex plane onto itself, generated by V(z)= — 1/z and U(z) = z+λq, where λq = 2 cos (π/q), qbeing a positive integer greater than 2. In this paper we shall be concerned only with the group given by q = 5, and we shall therefore omit the subscript 5 on the λ. We note that λ = λ5 satisfies the equation

x2–x–l=0;(1)

henceλ = (l + 5½)/2.

In this note I show that

where Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.

Let C denote Cayley's algebra defined over the field of rational numbers. This paper contains a simple characterization of arithmetics of C in terms of a given basis i0 = 1, i1, i2, …, i7. We deduce that certain of the arithmetics of C are isomorphic. The result that the maximal arithmetics are isomorphic is also contained in the work of van der Blij and Springer [2].

In this note a theorem, giving a relation between the Hankel transform of f(x) and Meijer's Bessel function transform of f(x)g(x), is proved. Some corollaries, obtained by specializing the function g(x), are stated as theorems. These theorems are further illustrated by certain suitable examples in which certain integrals involving products of Bessel functions or of Gauss's hypergeometric function and Appell's hypergeometric function are evaluated. Throughout this note we use the following notations:

The integral function

is known as Airy's Integral since, when z is real, it is equal to the integral

which first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.

Dr V. O. Ennola has pointed out that there is a mistake in this paper. The inequality for dox at the foot of page 76 does not imply the stated inequality between the second and third lines on page 77. This error vitiates the entire argument, and I have not so far been able to put it right.