In this paper we first of all solve the dual series equations

where ƒ(ρ) and g(ρ) are prescribed functions,

is the Jacobi polynomial (2).

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the form

Further, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

Stirling numbers of the first and second kind play an important part in many branches of mathematics, in particular in combinatorial analysis and statistics. For their definition and properties we refer to (5) where a whole chapter is devoted to their study. Stirling numbers have been generalized in many ways. One generalization is given in (1). In this paper we generalize the results of (1) to n dimensions. In order to simplify the notation we use methods of linear algebra.

Let ƒ(x) be a non-linear polynomial with rational integer coefficients, and for integral x let P(x) denote the greatest (positive) prime factor of ƒ(x). Pólya (1) has proved that if ƒ(x) is of degree 2 and has distinct roots then P(x)→∞ as x→∞. It is probably well-known that, provided ƒ(x) has distinct roots, this is true whatever the degree of ƒ(x). There does not appear to be a proof of this in the literature, but it is easily deducible from a result of Siegel (2). These results, however, are non-effective, although effective results have been obtained for a number of special polynomials. Chowla (3) has proved that, if ƒ(x) = x2 +1, then P(x)>C log log x, where C is an absolute positive constant. Analogous results have been proved for some polynomials of the form ax2+b and for some of the form ax3 + b by Mahler and Nagell respectively (4).

Let

The asymptotic behaviours of ƒ(x) and g(x), as x→+0, were first given by G. H. Hardy in (4), (5). In his papers {an}; is a monotone decreasing sequence. Further results on the asymptotic behaviours of ƒ(x) and g(x), as x→+0, for monotone coefficients have been given in (9) and (1). Recently, the results have been generalized to quasi-monotone coefficients.

This paper is concerned with asymptotic behaviours of ƒ(x) and g(x) for δ-quasi-monotone coefficients.

In what follows, we shall denote by L(x) a slowly varying function in the sense of Karamata (6), i.e.

When a function ƒ(x) possesses an asymptotic series

this series provides a useful means of evaluating ƒ(x) for large values of x. The usual procedure is to sum all the terms in S(x) up to, but excluding, the term of smallest magnitude. The degree of accuracy obtained by this method cannot normally be improved by direct summation of S(x), but sometimes better accuracy can be obtained by using one of the familiar devices for accelerating the convergence of series. Simple δ2-extrapolation may be successful, and Rosser (1) and others have used the Euler transformation to some effect. The method given here provides, in suitable cases, a more effective means of evaluating ƒ(x) from the series for a wide range of values of x.

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).

I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

The equation

need not have a solution z in the complex plane, even when ƒ is entire. For example, let ƒ(z) = ez, z1 = z0+2kπi. Thus the classical mean value theorem does not extend to the complex plane. McLeod has shown (2) that if ƒ is analytic on the segment joining z1 and z0, then there are points w1 and w2 on the segment such that where

The purpose of this article is to give a local mean value theorem in the complex plane. We show that there is at least one point z satisfying (1), which we will call a mean value point, near z1 and z0 but not necessarily on the segment joining them, provided z1 and z0 are sufficiently close. The proof uses Rouché's Theorem (1).

A semigroup S is said to be normal if aS = Sa for each a in S. Thus the class of normal semigroups includes the class of groups and the class of Abelian semigroups. Given a compact semigroup S we write P(S) for the convolution semigroup of probability regular Borel measures on S. In (3), Theorem 7, Lin asserts that a compact semigroup S is normal if and only if P(S) is normal. We show in this paper that Lin's result is false. In fact, if S is the union of subsemigroups each of which has an identity element, we show that P(S) is normal if and only if S is Abelian. Thus any compact non-Abelian group contradicts Lin's result. What Lin's argument does establish is that if P(S) is normal then S is normal, and if S is normal then μP(S) = P(S)μ for each point mass measure μ.

Let Vk denote the class of functions

which map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.

In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.

In this paper, the author continues his investigation, initiated in (4) and (5), into the nature of certain “arithmetical” functions associated with the factorisation of normalised non-zero polynomials in the ring GF[q, X1, …, Xk], where k ≧ 1, GF(q) is the finite field of order q and X1, …, Xk are indeterminates. By normalised polynomials we mean that exactly one polynomial has been selected from equivalence classes with respect to multiplication by non-zero elements of GF(q). With this normalisation GF[q, X1, …, Xk] becomes a unique factorisation domain. The constant polynomial will be denoted by 1. By the degree of a polynomial A in GF[q, X1, …, Xk], we shall mean the ordered set (m1, …, mk), where mi is the degree of A in Xi, 1 ≦ i ≦.k. We shall assume that A(≠ 1), a typical polynomial in GF[q, X1, … Xk], has prime factorisation

where P1, …, Pr are distinct irreducible polynomials (i.e. primes).