ABSTRACT
Functions
7.1 Introduction
Analogous to the generalizations of the family of gamma and beta functions,
extensions of the Riemann zeta function are presented, for which the usual
properties and representations are naturally and simply extended. Analo-
gous to these extensions, the extended Hurwitz zeta functions are introduced.
Hurwitz-type formulae are also proved. Because there are innitely many
new functions or generalizations possible for well-known functions, one needs
some clear-cut criteria to determine whether a given extension is worthwhile
or not. If it arises in diverse problems or interesting relations turn up between
it and other functions, or new insights are provided for the original functions,
or particularly elegant results can be found for the new function, it would be
worthwhile. In the present case, we do nd natural extensions of the previous
results, obtain new results for the extensions and expect that there will be a
wider applicability of these extended zeta functions. We use integral repre-
sentations of the zeta function to extend them analogously to the generalized
gamma function. We also apply our extension procedure to the Hurwitz zeta
function, (; q), and obtain the corresponding extensions for it. The asymp-
totic representations for these functions for small values of the parameter are
proved as well. Hurwitz-type formulae for the corresponding loop integrals
are also proved. In the rst three sections, we present standard results about
the Bernoulli numbers and polynomials and the Riemann zeta functions and
its zeros. We state the standard results about the Hurwitz zeta function in
Section 7.4. The rest of the sections deal with the extended Riemann and
Hurwitz zeta functions.
