| The case for t = 1 of the computational problems of the finite sum of the form:could be generally treated by means of well-known Euler-maclaurin summation formula, but few studies were available about (1) in case of t≠1. In 1995, G.F.C.DeBruyn[1]did research on a kind of arithmetic-geometry series as follows:Later, in 1999 and 2000, L.C.Hsh & P.J.-S.Shiue[2] and L.C.Hsh & E.L.Tan[3] respectively did several extends on DeBruyn's result. Before this, in 1998, professor Wang Xing-hua[4] found out a general formula to (1), which doesn't use the Eulerian function, but a kind of new combinatorial tools.Firstly, the paper prove two theorems about higher differentiable function represented by Bernoulli function or Eulerian function by the same way. Based on the theorems, we discuss their application in the summation problem of series including (1) and some analysis of the efficiency of fast algorithms and parallel algorithms.The paper is organized by three chapters.In chapter 1, based on generating function and calculation of Blissard, two theorems of higher differential functions represented by the Bernoulli number , the Bernoulli polynomial and the Bernoulli function or the Eulerian number , the Eulerian polynomial and the Eulerian function are given in this paper.Theorem 1 Let f(z) be a (n+l)-th continuous differentiable function within some intervals involving z , then the following representation is right:where f(-1)(z) represents the indefinite integral of f(z) , Bk(k = 0,1,2,…n+1) and B*n+1 (z) are Bernoulli numbers and Bernoulli function respectively.Theorem 2 Let f(z) be a (n+l)-th continuous differentiable function within some interval involving z, then the following representation is right:where t≠1,a*k(z,t) are Eulerian functions, ak(t}=, Ak(t)are Eulerianpolynomials.In particular, when f(z) is an n-th polynomial, two representations can be givenby Bernoulli function or Eulerian function without complement.The results above have many important applications in analyzing the efficiency( complexity ) of fast algorithms and parallel algorithms designed by half-cutting of recurrence technology . Therefore, in chapter 2, we did some researches about the general solutions of the following difference equation:when the situations of p = q and p ≠ q, where x > 0, z = loga x, T(x) is unknownfunction, f(z) is the (n+l)-th continuous differentiable function within some intervals involving z. The main results are the follow theorems.Theorem 3 Let f(z) be a (n+l)-th continuous differentiable function within some intervals involving z, then the following representation is right.where z = loga x, f(-1) (z) represents the indefinite integral of f(z) , are Bernoulli numbers and Bernoulli functions respectively, and C(x) represents an arbitrary function on (0,+∞), which satisfiesC(x) = C(x/a). Theorem 4 Let f(z) be a (n+l)-th continuous differentiable function within some intervals involving z, then the following representation is right.where z = logax, t = aq-p , ak*(z,k) are Eulerian functions, , andAk (t) are Eulerian polynomials.Equation (2) can be explained as follows: a problem of x -scaled can betransformed to a same problem of ap -s x/a scaled with a calculating overhead ofxq f(z) , so T(x) , the cost (complexity) of this problem, satisfies the above recursiveeqution.In chapter 3, we discusses the applications of the theorems in chapter 1 insummation problem. The main results are as follows:Theorem 5 Let f(z) be a (n+l)-th continuous differentiable function within some intervals involving z , then the following representation is right:where N is positive integer, f(-1)(z) represents the indefinite integral of f(z) , are Bernoulli numbers and Bernoulli functions respectively.Theorem 6 Let f(z) be a (n+1)-th continuous differentiable function withinsome intervals involving z , then the following representation is right when t ≠0,1 :where N is positive integer a*k(z,k) are Eulerian fun... |
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