Asymptotic expansions of the Hurwitz–Lerch zeta function

Continued-fraction Expansions for the Riemann Zeta Function and Polylogarithms

It appears that the only known representations for the Riemann zeta function ζ(z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ζ(n) for any integer n ≥ 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n ≥ 1, by using the classical Stieltjes technique. Our...

Asymptotic Expansions

Asymptotic expansions of functions are useful in statistics in three main ways. Firstly, conventional asymptotic expansions of special functions are useful for approximate computation of integrals arising in statistical calculations. An example given below is the use of Stirling's approximation to the gamma function. Second, asymptotic expansions of density or distribution functions of estimato...

On some expansions for the Euler Gamma function and the Riemann Zeta function

Abstract In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of con...

Asymptotic Expansions Methodsand Applications

Feynman diagrams are the most important theoretical tool for particle physicists. They are an efficient link between theory and experiment. However, their translation into actual numerical predictions is often very tedious if not impossible. Huge efforts have been devoted to their evaluation, and several powerful methods have been developed to systemize their treatment. The more complex a Feynm...

Asymptotic Expansions for Integrals