Anharmonic polynomial generalizations of the numbers of Bernoulli and Euler

Abstract

We consider twelve infinite systems of polynomials in z which for z = 1 degenerate either to the numbers of Bernoulli or Euler, or to others simply dependent upon these. The first part proceeds from the definition of anharmonic polynomials to the specific twelve systems discussed; the second presents an adaptation of the symbolic calculus of Blissard and Lucas in sufficient detail for rapidly developing a simple isomorphism between the algebra of the polynomials and that of the twelve elliptic functions sn, cn, ns, nc, sc, ... of Glaisher, and the third contains a short selection from the simpler algebraic and congruential relations between the polynomials. Incidentally there is pointed out in the second part a new interpretation of Kronecker's work on certain symmetric functions and their connections with Bernoulli's nlumbers. Owing to the length of the paper the development stops short of the quadratic transformation of the polynomials which corresponds to the transformation of the second order in elliptic functions, but the material given is a necessary foundation for all higher transformations. For the same reason only prime moduli are considered in the congruences, although the case in which the modulus is a power of a prime can be treated in essentially the same way, but at greater length. All references are at the end of the paper.