The Laguerre polynomials Ln(α)(x) have the following hypergeometric and Rodrigues representation (3, pp. 188, 189):

1,

2.

In a recent paper (4) Touchard has constructed a set of polynomials Ωn(z) such that

(1),

where after expansion of the left member Bm is replaced by Bm,

,

and

(2).

Wyman and Moser (5) have recently given an explicit form for some polynomials of Touchard (4). Equation (6) of this note will give a simpler expression for these polynomials. Subsequently, generating functions (9) and (12) will be obtained.

In a previous paper (1) the authors considered the problem of finding an asymptotic formula for numbers or functions Bn,m whose generating function is of the form

(1.1),

where Pm(x) is a polynomial of degree m in x given by

(1.2), am≠0.

The above-mentioned paper contained the restriction that ak ≥ 0.

The network-flow problem, originally posed by T. Harris of the Rand Corporation, has been discussed from various viewpoints in (1; 2; 7; 16). The problem arises naturally in the study of transportation networks; it may be stated in the following way. One is given a network of directed arcs and nodes with two distinguished nodes, called source and sink, respectively. All other nodes are called intermediate. Each directed arc in the network has associated with it a nonnegative integer, its flow capacity. Source arcs may be assumed to be directed away from the source, sink arcs into the sink. Subject to the conditions that the flow in an arc is in the direction of the arc and does not exceed its capacity, and that the total flow into any intermediate node is equal to the flow out of it, it is desired to find a maximal flow from source to sink in the network, i.e., a flow which maximizes the sum of the flows in source (or sink) arcs.

Thus, if we let P1 be the source, Pn the sink, we are required to find xij (i,j =1, . . . , w) which maximize

If a1, a2, … , ak are relatively prime positive integers then the equation

always has solutions in non-negative xi for n sufficiently large. Sylvester called the number of non-negative solutions of (1) the denumerant of the equation. We shall denote the denumerant by n(a1, … , ak).

Suppose αn ≠ 0 for all n. By replacing a set of terms from the series Σαn by 0's, we form a subseries of the original series. These subseries can be put in one to one correspondence with the non-terminating binary expansion of the points on the real line segment (0, 1).

Heawood (3) showed that for h > 1 the countries of any map on a surface of connectivity h = 3 − x can be coloured using at most

colours. In a previous paper (1)1 proved the following

Improving on inequalities given by Gerretsen (2), Beatty (1) has proved that for the area Δ of any plane triangle with sides a, b, c the following inequalities hold:

(1.1)

where

(1.2);

the signs of equality in (1.1) only apply when the triangle is equilateral. Beatty has also remarked that the second inequality in (1.1) is of no value in case 5H ≥ 3K, since then the lower estimate which it gives for Δ2 is not even positive.

A purely algebraic solution for the problem of the extreme definite ternary quadratic form has been given by Mordell (2). The equivalent geometric problem of the critical lattice of a sphere was solved by Mrs. Ollerenshaw (3) under a preliminary assumption that was said to be justified by “simple considerations” (p. 297).

This paper is devoted partly to a summary of some results which will be published in detail elsewhere, partly to a description of a new extreme senary form, ϕ6, and an n-dimensional generalization of it.

In complex affine n-space with a unitary metric, a reflection is a congruent transformation leaving invariant all the points of a hyperplane. Thus the characteristic roots of a unitary reflection of period p consist of a primitive pth root of unity and n – 1 unities. A group generated by n reflections is conveniently represented by a graph having a node for each generator and a branch for each pair of non-commutative generators.

This paper is an appendix to “Finite unitary reflection groups” by J. A. Todd and the author (3) in which the irreducible finite groups generated by reflections in unitary space (2, p. 364) are enumerated and abstract definitions are given for all those n-dimensional groups which may be generated by n reflections.

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality

(1)

has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

Throughout this note all groups are abelian, written additively. We refer to Kurosh (8; 9) for notation, terminology and theorems used without reference. We recall the notion of a serving subgroup (or pure subgroup) of a group . This is a subgroup in which for every natural number n every equation nx = s, s ∊ can be solved provided that it can be solved in . If is torsion-free, “linearly closed” subgroups coincide with serving subgroups and is a serving subgroup if and only if / is torsionfree.

Let A, B be two square matrices with complex coefficients, of respective orders n and m, where n ≥ m. We shall say that B is imbeddable in A if there exists a unitary matrix U of order n such that U*AU contains B a s a principal submatrix. In other words, B is said to be imbeddable in A if there exists a matrix V of type n × m such that V*V = IW (= the identity matrix of order m) and V*AV = B.

The purpose of this paper is to present two concavity results for symmetric functions and apply these to obtain inequalities connecting the characteristic roots of the non-negative Hermitian (n.n.h.) matrices A, B and A + B.

In 1951 Fan (1) proved the following interesting extreme value result: Let A1, … , Am be completely continuous operators on a Hilbert space . For be the characteristic roots of Aσ* Aσ. Then, for any positive integer k,

(1),

(2),

where both maxima are taken over all unitary operators U1, … , Um and all sets of k orthonormal (o.n.) vectors.

1. Introduction. Burnside's Theorem in the theory of group representations states that a necessary and sufficient condition that a semigroup of matrices of degree n over the complex field be irreducible is that the semigroup contain n2 linearly independent matrices. In the course of dealing with sets of matrices with coefficients in a division ring, Brauer (1) obtained a generalization of this theorem which concerned irreducible semigroups with elements in a division ring.