The text is concerned with the definition and investigation of some classes of ordinal valued-functions; and with the determining of a particular assignment function O. This function assigns to each limit ordinal alpha smaller than a constant theta and belonging to the second number class a fundamental sequence [diagram] that is, a strictly increasing sequence satisfying [diagram]. The operations used to determine O are derived from a class of functions [diagram]. Since generalizes the standard ordinal arithmetic operations,some of the properties of the are studied, together with the operations gammaalpha which are generalizationsof transfinite sum and product. Also investigated is a related class of number-theoretic functions, in this context O is an arbitrary assignment bounded by o in place of [diagram]. It is proved that the functions are normal in the second argument, and these functions are compared with a hierarchy of normal functions obtained by Veblen's process of iteration. The provide a natural means of extending the notion of epsilon number, and some of the properties of the generalized epsilon numbers are presented. The function of [diagram] is normal, and the notation [diagram] is adopted for the sequence of countable fixed points of the function. The generalized epsilon numbers determine a hierarchical classification of limit numbers, and on this basis a normal form is determined for each, and thence the function O is defined by transfinite recursion.