Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums

We show that integrals of the form ∫ 1 0 xLip(x)Liq(x)dx (m ≥ −2, p, q ≥ 1) and ∫ 1 0 log(x)Lip(x)Liq(x) x dx (p, q, r ≥ 1) satisfy certain recurrence relations which allow us to write them in terms of Euler sums. From this we prove that, in the first case for all m, p, q and in the second case when p+ q+ r is even, these integrals are reducible to zeta values. In the case of odd p+q+r, we comb...

Double Shuffle Relations of Euler Sums

Abstract. In this paper we shall develop a theory of (extended) double shuffle relations of Euler sums which generalizes that of multiple zeta values (see Ihara, Kaneko and Zagier, Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142 (2)(2006), 307–338). After setting up the general framework we provide some numerical evidence for our two main conjectures. At the ...

Many-Electron Integrals over Gaussian Basis Functions. I. Recurrence Relations for Three-Electron Integrals.

Explicitly correlated F12 methods are becoming the first choice for high-accuracy molecular orbital calculations and can often achieve chemical accuracy with relatively small Gaussian basis sets. In most calculations, the many three- and four-electron integrals that formally appear in the theory are avoided through judicious use of resolutions of the identity (RI). However, for the intrinsic ac...

Recurrence relations for quantal multipole radial integrals

In a previous paper, we showed that in the semiclassical (WKB) Coulomb approximation, the radial integrals for mu)tipole transitions between nonhydrogenic atomic states obey simple recurrence relations. This paper deals with the recurrence relations connecting the nonhydrogenic radial matrix elements expressed in the quantal form. Numerical tests have been made and give good results.

Recurrence Relations and Generating Functions