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Simultaneous polynomial approximations of the Lerch function. (English) Zbl 1186.41006
This article is concerned with the polynomial approximation of the Lerch function \(\Phi_s(x,z)=\sum_{n=1}^\infty z^n(n+x)^{-s}\). This definition includes the Hurwitz function \(\zeta(s,z)=\Phi_s(x,1)\), the polylogarithms \(Li_s(z)=\Phi_s(0,z)\), and the Riemann zeta function \(\zeta(s)=\Phi_s(0,1)\), all of which have been studied with different intensities. The aim of this work is to find explicit formulae for certain diagonal Hermite-Padé type approximants of the function \(\Phi_s(x,z)\). One of the problems considered and solved by the author is the following:
Given integers \(A\geq 2\), \(n\geq 0\), \(r \geq 0\) such that \(A(n+1)\geq r+2\), find \(A+1\) polynomials \(P_0(x,z), P_1(x,z),\dots,P_A(x,z)\) in \(\mathcal{Q}[z,x]\), of degree at most \(r\) in \(x\) and at most \(n\) in \(z\), and \(\widehat{P}_0(x,z)\in \mathcal{Q}(x)[z]\) of degree at most \(n\) in \(z\), such that \(P_0(x,z)+\sum_{j=1}^{A} P_j(x,z)\Phi_j(x,1/z)=\mathcal{O}(x^{-A(n+1)+r+1})\) at \(x=+\infty\), and \(\widehat{P}_0(x,z)+\sum_{j=1}^{A} P_j(x+r,z)\Phi_j(x,1/z)=\mathcal{O}(z^{-r-1})\) at \(z=+\infty\).
This theory includes known results on polylogarithms, and also generalises recent works of Beukers and Prévost related to Hurwitz zeta functions.

11J72 Irrationality; linear independence over a field
11M35 Hurwitz and Lerch zeta functions
11M32 Multiple Dirichlet series and zeta functions and multizeta values
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A21 Padé approximation
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