In this paper two integrals involving E-functions are evaluated in terms of E-functions. The formulae to be established are:

where n is a positive integer,

and

where n is a positive integer,

and

the prime and the asterisk denoting that the factor sin {(s–s)π/2n} and the parameter βq+s–βq+s + 1 are omitted. The definitions and properties of MacRobert's E-function can be found in [1, pp. 348–352] and [3, pp. 203–206].

For any compact convex set K ⊂ ℂ there is a unital Banach algebra Ea(K) generated by an element h in which every polynomial in h attains its maximum norm over all Banach algebras subject to the numerical range V(h) being contained in K, [1]. In the case of K a line segment in ℝ, we show here that Ea(K) does not have Arens regular multiplication. We also show that ideas about Ea(K) give simple proofs of, and extend, two inequalities of C. Frappier [4].

This paper is paper gives what appears to be a new Rodrigues’ formula for the Associated Legendre Polynomials defined by [5, p. 122]

with the restriction that m is an even positive integer, which helps to evaluate some integrals. Putting m = 2k in (1.1) and replacing Pn(x) by the Gegenbauer Polynomial and using [3, p. 176]

We obtain

Putting a =v–½ in the relation [4, p. 283]

We get

An important step in the proof of Kostrikin's fundamental theorem [2] on finite groups of prime exponent is the following result.

Theorem 1. Let L be a Lie algebra of characteristic p satisfying the t-th Engel condition for some t < p, and suppose that L is generated by elements that are right-Engel of length 2. Then L is locally nilpotent.

In integrating E-functions with respect to their parameters the contours are usually of the Barnes type, deformed if necessary to separate the increasing and decreasing sequences of poles of the integrands. Also the constants are taken to be such that the integrals converge. The following formulae are required in proving the theorems given in this paper.

Peter May kindly tells me that the proof of the Nishida relations in §8 of this paper works only for E∞ spaces, not for H∞ ring spectra. The reason is that there is no suitable “diagonal” map d for H∞ ring spectra. The result is correct all the same (it is not mine), and its formulation in §3 is also correct.

If G is the group of holomorphic automorphisms of a bounded symmetric domain, then G has a distinguished class of irreducible unitary representations called the holomorphic discrete series of G. These representations have been studied by Harish-Chandra in [7]. On the Lie algebra level, the Harish-Chandra modules corresponding to the holomorphic discrete series representations are highest weight modules. Even for G as above, it turns out that not all the unitary highest weight modules belong to the holomorphic discrete series but there exists a condition on the highest weight which characterizes the holomorphic discrete series among the unitary highest weight representations. They can be defined as those unitary highest weight representations with square integrable matrix coefficients.

Let G be a given group and A, B be two subgroups of G which may or may not coincide. A homomorphism μ which maps A onto B is called a partial endomorphism of G. When A coincides with G then we call μ a total endomorphism or as it is usually called an endomorphism of G. If μ* is a partial (or total) endomorphism of a supergroup G* ⊇ G, then we say that μ* extends, or continues, μ when μ* is defined for at least all the elements a ∈ A and moreover aμ = aμ* for all a ∈ A If the partial endomorphism μ is an isomorphic mapping then we speak of a partial automorphism of G.

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.

The Wielandt subgroup ω(G) of a group G is defined to be the intersection of all normalizers of subnormal subgroups of G; the terms of the Wielandt series of G are defined, inductively, by putting ω0(G) = 1 and (ωn+1(G)/ωn(G) = ω(G/ωn(G)). If, for some integer n, ωn(G) = G, then G is said to have finite Wielandt length; the Wielandt length of G being the minimal n such that ωn(G) = G.

Recently several authors have studied dualizing Goldie dimension of a module: spanning dimension in [2], codimension in [13], corank in [16] and also [9,17,12, 5,11, 6, 4, 7] ([13] may be read in comparison with the others). In the present note we prove the equality corank RP = corank SS, where P is a quasi-projective left R-module and S is its endomorphism ring. This result is an answer to the question [12, p. 1898] and an extension of [3, Corollary 4.3] which shows the above equality for a Σ-quasi-projective left R-module P.

Let n = be the factorization of an integer n(>1) into prime powers, and set Φ(n):= . In particular, for squarefree n, Φ(n) = phi;(n). Consider the set

.

It is known (from [5]) that A consists precisely of those integers n for which there is no non-abelian group of order n. It is also known (from [7]) that the set

consists solely of integers n with the property that every group of order n is cyclic. We set C′ = A – C.

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.

In this note we give the proof of the following result (previously known for homotopically trivial and free actions on infranilmanifolds [3, Theorem 5.6]).

Theorem 1. Let G be a finite group acting freely and smoothly on a closed infranilmanifold M. Assume that dim M≠3, 4. Then the action of G is topologically conjugate to an affine action.

Let R be a commutative ring (with non-zero identity) and let M be an R-module.

Let G be a polycyclic-by-finite group and let K[G] denote its group algebra over the field K. In this paper we discuss localization in K[G] and in particular we prove that every faithful completely prime ideal is localizable. Furthermore, using a sequence of localizations, we show that, for G polyinfinite cyclic, the classical right quotient ring (K[G]) is in fact a universal field of fractions for K[G]. Finally we offer an example of a domain K[G] which does not have a universal field of fractions.

An associative ring R is called a left SI-ring if every singular left R-module is injective. In Goodearl [4] it is shown that these rings have a finite ring decomposition into a ring K with K/Soc K left semisimple, and simple rings which are Morita equivalent to left SI-domains.

A general theory of Hankel forms over domains in one or several variables has been set forth in [6]. In [7] the study of Hankel forms over an annulus in the complex plane ℂ was begun. (An extension of the results of [7] to multiply connected domains was given in [4].) The present paper amplifies the results of [7] in various respects. First of all we define and study more general Hankel forms associated with a one parameter family of projective structures on the annulus. This displays several new features. For instance, we are now dealing with quadratic integral metrics which do not correspond to integration of the square of the function with respect to a weight. Furthermore, whereas in [7] essentially only the issue of the boundedness of Hankel forms was studied, we obtain here rather satisfactory Sp-results, even for 0 < p < 1. The question which remains is, of course, to which extent all this extends to multiply connected domains (or more general (open) Riemann surfaces).