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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.

##### MSC:
 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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