Riemann \(\mathfrak P\)-scheme, monodromy and Diophantine approximations (Q441373)

This paper investigates the Padé approximations of the first and second kind for the Lerch functions defined by NEWLINE\[NEWLINE\Phi_k(x,z)=\sum_{n=1}^\infty {(1/z)^n\over(n+x)^k}.NEWLINE\]NEWLINE If the non-negative integers \(n, \sigma_0,\sigma_1,\sigma_\infty\) satisfy \(\sigma_\infty+\sigma_0+\sigma_1=q(n+1)\), then there are polynomials \(A_k(z)\) for \(0\leq k\leq q\) of degree at most \(n\) and not all trivial such that NEWLINE\[NEWLINE\begin{align*}{R_\infty(x,z)&=A_0(z)+\sum_{k=1}^q A_k(z)\Phi_k(x,z)=O(1/z)^{\sigma_\infty}\cr R_0(x,z)&=A_0(z)+\sum_{k=1}^q (-1)^{k+1}A_k(z) \bigg(\Phi_k(-x,1/z)+{(-1)^k\over x^k}\bigg)=O(z)^{\sigma_0}\cr R_1(x,z)&=\sum_{k=1}^q {A_k(z)\over (k-1)!}z^x\log(1/z)^{k-1}=O(z-1)^{\sigma_1}\cr}\end{align*}.NEWLINE\]NEWLINENEWLINENEWLINEThe polynomials \(A_k(z)\) are determined uniquely up to a constant factor. With the normalisation \(A_q(0)=1\), they can be given explicitly in terms of hypergeometric functions. Lerch's function is the holomorphic solution of the Fuchsian differential equation \(\mathcal A\) of order \(k+1\) NEWLINE\[NEWLINE{\mathcal A}: (\theta(\theta-x)^k-z(\theta+1)(\theta-x)^k)\Phi_k(x,z)=0.NEWLINE\]NEWLINE The conditions for \(R_0\) and \(R_1\) arise from the analytic continuations of \(R_\infty\) at 0 and 1 respectively. The explicit representation of the polynomials and reminders in terms of hypergeometric functions is obtained from the general Riemann theory of such systems of differential equations and hypergeometric identities.NEWLINENEWLINEThe Hurwitz zeta function is the special case \(\zeta(k,x)=\Phi_k(x,1)\). The general theory yields a new construction for simultaneous rational approximations for linear combinations of the \(\Phi_k(x,z)\) for \(p\leq k\leq p+q\). This leads to diophantine approximations for \(\zeta(3,x)\) obtained in another way by \textit{F. Beukers} [Acta Math. Sin., Engl. Ser. 24, No. 4, 663--686 (2008; Zbl 1169.11029)] and also Apéry's simultaneous rational approximations for \(\zeta(2)\) and \(\zeta(3)\).NEWLINENEWLINEThe polylogarithm is the special case \(Li_k(1/z)=\Phi_0(0,z)\) and \(Li_k(-1)=\big(1-{1\over2^{k-1}}\big)\zeta(k)\). The explicit Padé approximants of the second kind for the family \(\Phi_1(x,z),\ldots,\Phi_k(x,z)\) lead to \textit{T. Rivoal}'s formulas [Can. J. Math. 61, No. 6, 1341--1356 (2009; Zbl 1186.41006)] for simultaneous asymptotic expansion of \(\zeta(2,x),\ldots,\zeta(q,x)\) and so to simultaneous rational approximations of \(1, \log2, \zeta(2),\ldots,\zeta(q)\), as well as other diophantine approximations.