If p ≧ q + 1, [1, p. 353]

where, if p = q + 1, | z | < 1. The dash in the product sign indicates that the factor for which t = αr – αr + 1 is omitted, while the asterisk indicates that the parameter αr−αr + 1 is omitted.

Since [3]

where , a result involving Wk, m, (z) can be transformed into a result involving MacRobert's E-function. Further this result can be generalised with the help of the known integrals for E-functions.

The object of this paper is to use this method to obtain some recurrence relations and series for MacRobert's E-functions.

A function φ(p) is operationally related to h(t) when they satisfy the integral equation

provided that the integral is convergent and R(p)> 0.

As usual, we shall denote (1) by the symbolic expression

φ(p) ≑ h(t).

The object of this paper is to evaluate some infinite integrals involving E-functions by the methods of the operational calculus. Most of the results obtained are believed to be new.

By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quartic

of genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(g − 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper.

In previous papers [1, 2, 3] the sums of a number of series of products of E-functions have been found. For the definitions and properties of the E-functions the reader is referred to [4, pp. 348–358]. In § 3 a further series of this type is given. The proof is based on an integral of an E-function with respect to its parameters, to be established in § 2. Similar integrals were given in [5] and [6].

1. Suppose throughout that a, k are positive numbers and that p is the integer such that k—l≦p<k. Suppose also that φ(w), ψ(w) are functions with absolutely continuous (p+1)th derivatives in every interval [a, W] and that φ(w) is positive and unboundedly increasing. Let λ ={λn} be an unboundedly increasing sequence with λ1 > 0.

Given a series and a number m ≧0, we write

otherwise,

and A(w) = AO(w).

For any positive integers n and v let

where d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,

as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identity

where x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formula

are of interest in number theory and combinatory analysis. He proves the apparently new and striking result

where |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.

Let E be an arbitrary (non-empty) set and S the restricted symmetric group on E, that is the group of all permutations of E which keep all but a finite number of elements of E fixed. If Φ is any commutative ring with unit element, let Γ = Φ(S) be the group algebra of S over Φ,Γ ⊃ Φ and let M be the free Φ-module having E as Φ-base. The “natural” representation of S is obtained by turning M into a Γ-module in the obvious manner, namely by writing for α∈S, λ1∈Φ,

By a ‘representation’ we shall mean throughout a representation by n × n matrices with entries from an arbitrary field. Elsewhere [9] the author has introduced the concept of a principal representation of a semigroup S (see § 3 below for the definition) and has shown that if S satisfies the minimal condition on principal ideals then every irreducible representation is of this type. Moreover, if S satisfies the minimal conditions on both principal left and right ideals, which together imply the minimal condition on principal two-sided ideals [6, Theorem 4], the irreducible representations of S can ultimately be expressed explicitly in terms of group representations.

The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.

In the analysis of mixed boundary value problems in the plane, we encounter dual integral equations of the type

If we make the substitutions cos we obtain a pair of dual integral equations of the Titchmarsh type [1, p. 334] with α = − 1, v = − ½ (in Titchmarsh's notation). This is a particular case which is not covered by Busbridge's general solution [2], so that special methods have to be derived for the solution.