Analytic continuation of Jacobi polynomial expansions

Analytic continuation and perturbative expansions in QCD

Starting from the divergence pattern of perturbative quantum chromodynamics, we propose a novel, non-power series replacing the standard expansion in powers of the renormalized coupling constant a. The coefficients of the new expansion are calculable at each finite order from the Feynman diagrams, while the expansion functions, denoted as Wn(a), are defined by analytic continuation in the Borel...

Product Jacobi Expansions 3

Polynomial Expansions

The expansion of arbitrary power series in various classes of polynomial sets is considered. Some applications are also given. Notations. We will use the following contracted notation for the generalized hypergeometric function •i> •flB P q\bl,...,bQ pFAb, ^ M* z_k where p. « r(o4 k) («iOfc-n fy)*. (*v*s rw* and (°)* / = ! / = ! r(o)

On Effective Analytic Continuation

Until now, the area of symbolic computation has mainly focused on the manipulation of algebraic expressions. It would be interesting to apply a similar spirit of “exact computations” to the field of mathematical analysis. One important step for such a project is the ability to compute with computable complex numbers and computable analytic functions. Such computations include effective analytic...

Analytic continuation of multiple polylogarithms

It is not hard to see that this function can be analytically continued to a multi-valued memomorhpic function on C. When n > 1 the special value Lin(1) is nothing else but the Riemann zeta value ζ(n), which has great importance in number theory. In recent years, there has been revival of interest in multi-valued classical polylogarithms and their single-valued cousins. For any positive integers...