The object of this note is to give an aspect to the problem of the functional equation of the generalized gamma function and Dirichlet series which are defined in [1]. In general, we cannot answer the problem yet. But it is worthy to attack this problem for some special cases.

Recently Brown [1] gave two new classes of generating functions which include the generating functions for the polynomials of Gegenbauer, Jacobi and Laguerre. The aim of the paper is to give a new class of generating functions which includes both sets of generating functions given by Brown and provides a new class of generating functions for the polynomials of Gegenbauer, Jacobi and Laguerre.

The concepts of order and type associated with an entire function over the complex plane admit different “natural” extensions, most of which are in vogue, for the case of an entire function over 𝒞k, the cartesian product of k copies of the complex plane. This work is concerned with the relations among such extended concepts and with the analogous properties of the concepts.

I establish here a criterion for a sequence of ordinals to be generic over a transitive model of ZFC with respect to a notation of forcing first considered by Prikry in his Doctoral dissertation [2]. In Section 0 I review some notation, in Section 1 I list some facts about measurable cardinals, and in Section 2, after giving Prikry's result, I state and prove mine.

Let S be a compact semitopological semigroup and let P(S) be the convolution semigroup of probability measures on S. An idempotent measure μ in P(S) is defined to be primitive if and only idempotent measures in μP(S)μ are μ and the zero element m of P(S). In a previous paper [2] we give some characterization of primitive idempotent measures on S. Let Π(P(S)) be the set of primitive idempotents in P(S) and let Πc be the set of central primitive idempotents in P(S). It is shown in [1] that Π(P(S)) is neither an ideal nor even a subsemigroup of P(S) in general. The purpose of this paper is to investigate the structure of Πc.

In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

Let k and r be fixed integers such that 1 < r < k. It is well-known that a positive integer is called r-free if it is not divisible by the r-th power of any integer > 1. We call a positive integer n, a (k, r)-integer, if n is of the form n = a kb, where a is a positive integer and b is a r-free integer. In the limiting case, when k becomes infinite, a (k, r)-integer becomes a r-free integer and so one might consider the (k, i) integers as generalized r-free integers.

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.

In part I, I considered the problem of discovering when, given an irrational α which has a simple continued fraction representation with convergents pn/qn, there exists α' for which the denominator sequence for convergents is a subsequence of (qn). It was shown that such an α' exists if the continued fraction representation was “nearly periodic” with odd period. The following is a generalization of the results of part I to semi-regular continued fractions, where the problem seems to fit more naturally.

Abstract: We show that a tree is stable if and only if its automorphism group contains a transposition. The method is constructive and so enables a stabilising sequence to be found.

Since 1954, when Yang and Mills [7] presented their idea of isotopic gauge transformation, the method of introducing interactions into field theories by using general gauge invariance has been extensively studied.

A general formalism was presented by Utiyama [6]. Reference [6] also contains the first application of the formalism to the theory of gravitation. A more general approach to the Young-Mills formalism applied to general relativity was described by Kibble [3]. In a special case of interacting Dirac field the gauge invariance group can still be enlarged, leading to the possibility of describing short-range interactions together with gravitation and electromagnetism [5]. It is, therefore important to have a definite formulation of the common geometrical content of such theories.

Suppose S is a semitopological semigroup. We consider various subspaces of C(S) and determine what topological algebraic structure can be introduced into the spaces of means on the subspaces and into the spectra of the C*-sub-algebras of C(S) they generate.

The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n. This note gives an expansion for the permanent of such a line segment as a weighted average of permanents of matrices in Kn. For a survey article on permanents the reader is referred to Marcus and Mine [3].