1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence

for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)

In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B

In a recent paper [1] of Bell, an abstract inversion principle has been formulated for inverting a type of finite series by employing operators. Bell's result involves Baker's general principle of cross-classification [2], Dedekind-Möbius inversion, L.C.M. inversion and some known generalisations. The purpose of this note is to introduce operators of negative degree and to formulate an inversion principle which covers more cases than Bell's.

The study of non-associative algebras led to the investigation of identities connecting powers of elements of such algebras. Thus Etherington1 (1941, 1949, 1951) introduced the concept of the logarithmetic of an algebra, defining it roughly as “ the arithmetic of the indices of the general element”.

An integer n may be partitioned into the set of integers (α1, α2, …, αl), where Whenever αi−1>αi the substitution of αi+1 for αi, where αi may be αi+1≡0, would give a partition of n+1 whilst the substitution of αi−1—1 for αi−1 would give a partition of n—1; partitions of n+1 and of n—1 so arising will be said to be associated with the given partition of n.

Let σk(x) be the number of integers n≤x which are the product of just k prime factors, so that

and let πk(x) be the number of such n for which all the pi are different.

Let xik (i, k=1, 2, …, n) be n2 independent variables, aik be n2 constants, and .

Let x=(xik) be the determinant formed from the n2 elements xik, and the determinantal operator formed from the elements .

1. Let [ast] (s, t=0, 1, … n) be a square matrix of order n+1 and determinant |ast| and suppose that by repeated “isolation” of the variables the corresponding bilinear form has been expressed as

where, for all r,

Then

Now (1) implies, and is implied by, the identities

Thus, from any known identity of the form (4), subject to the condition (2), we may at once infer, using (3). the value of the corresponding determinant |ars|.