A Summation Formula for Power Series Using Eulerian Fractions*

where / ( / ) is a differentiate function defined on the real number interval [a, b), and x may be a real or complex number with x*0 and x*l. Obviously, the case for x = l of (1.1) could be generally treated by means of the well-known Euler-Maclaurin summation formula. The object of this paper is to find a general summation formula for (1.1) that could be applied readily to some interesting special cases, e.g., those with f{i)-t (2 ei?), / ( / ) = logf (V>1), and f(t) = q (0<q<l)9 respectively. Related results will be presented in Sections 3-5. Recall that for the particular case f(t) = t and [a, b) = [0, oo) with/? being a positive integer, we have the classical result (cf. [1], [2], [4])