An algebra of sequences of functions, with an application to the Bernoullian functions

Abstract

On account of its simplicity, its immediate applicability to many functions already in existence, and its suggestiveness with regard to generalizations of known functions or the creation of new, the algebra in question merits the somewhat detailed exposition in Part I. As its use is perhaps best seen from an example, we have sketched briefly in Part II the outlines of a new theory of the relations between the Bernoullian and Eulerian functions of any arguments and any ranks, showing that by means of the algebra all necessary computations are reduced to a minimum. These functions are of two variables, of which one is complex and the other, the rank, a positive integer. The algebra establishes a simple isomorphism between the theory of relations between the functions and the like for the ordinary sine and cosine. (It is necessary here to distinguish between ordinary and umbral circular functions; the nature of the distinction appears presently.) In a paper which I expect to publish later there is a more detailed application of this algebra to certain new functions, suggested by the algebra, of three variables, of which two are complex and the third a positive integer. For unit values of one of the complex variables these functions degenerate to the Bernoullian and Eulerian functions of Part II; for unit values of the other, they become certain polynomials, of considerable importance in the arithmetical theory of quadratic forms, discussed on several occasions by Hermite, Weierstrass and, more recently, by Bulygain and Gruder. In this application there are simple isomorphisms with the circular, the hyperbolic and the elliptic functions. In a third paper, to appear shortly, I have shown how the algebra gives at once the complete theory of the relations between the functions of Spitzer, which include as special cases the Bessel coefficients. Here there is simple isomorphism with the exponential function. By a generalization of the exponential function, Emxm+;/r(m+v), in which x, v are complex, I have shown in a paper not yet published that the algebra is readily applicable to the Bessel functions. The algebra therefore is of considerable utility. Its chief use, how-