Rational approximations to values of Bell polynomials at points involving Euler's constant and zeta values (Q2898888)

Generalizing the construction from their paper [\textit{On a continued fraction expansion for Euler's constant} (Preprint, \url{http://arxiv.org/abs/1010.1420})], the authors present new simultaneous approximations converging subexponentially to the values of Bell polynomials at points of the form NEWLINE\[NEWLINE\left(\gamma,1!(2a+1)\zeta(2),2!\zeta(3),\ldots,(m-1)!(a+(-1)^ma)\zeta(m)\right)NEWLINE\]NEWLINE for \(m=1,2,\ldots,a\) and \(a\in\mathbf{N}\).NEWLINENEWLINE\vskip0.2cm To state the main results, we need the definition of the exponential Bell polynomials: the polynomials \(Y_n(x_1,\ldots, x_n)\) follow from the formal power series expansion NEWLINE\[NEWLINE\exp{\left(\sum_{m=1}^{\infty}\,x_m{t m\over m!}\right)}=\sum_{n=0}^{\infty}\,Y_n(x_1,\ldots,x_n){tn\over n!},NEWLINE\]NEWLINE with explicit representation NEWLINE\[NEWLINEY_n(x_1,\ldots,x_n)=\sum_{\pi(n)}\,{n!\over k_1!\cdots k_n!}\left({x_1\over 1!}\right)^{k_1}\cdots\left({x_n\over 1!}\right)^{k_n},NEWLINE\]NEWLINE where the summation is taken over all partitions \(\pi(n)\) of \(n\) into \(n\) non-negative integers \(k_j\) with NEWLINE\[NEWLINE\sum_{j=1}^n\,jk_j=n.NEWLINE\]NEWLINENEWLINENEWLINEThe main results are now:NEWLINENEWLINENEWLINE\textbf{Theorem 1.1} Let \(a\geq 2\) be an integer. For \(\mu=1,2,\ldots,a-1\) and any non-negative integer \(n\) define the sequences of rational numbers NEWLINE\[NEWLINEq_n:=\sum_{k=0}^n\,{n\choose k}^ak!\in\mathbf{Z},\;p_{n,\mu}:=\sum_{k=0}^n\,{n\choose k}^ak! Y_{\mu}(r_1(k),\ldots, r_{\mu}(k))\in\text\textbf{Q},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEr_m(k)=(m-1)! (aH_{n-k}^{(m)}+(-1)^mH_k^{(m)}),\;k=0,1,\ldots,n.NEWLINE\]NEWLINE [\(H_n^{(m)}=\sum_{k=1}^n\,1/k^m,\;m\geq 1; H_0=0.\)]NEWLINENEWLINENEWLINELet NEWLINE\[NEWLINE\alpha_{\mu}:=Y_{\mu}\left(\gamma,1!(2a-1)\zeta(2),2!\zeta(3),\ldots,(\mu-1)!(a+(-1)^{\mu}(a-1))\zeta(\mu)\right).NEWLINE\]NEWLINENEWLINENEWLINESuppose that the coefficients \(b_m(a)\) are defined by NEWLINE\[NEWLINE-a\log{\left(1+\sum_{m=1}^a\,{(2-{m+1\over a})_m\over (m+1)!}z^m\right)}-\sum_{m=1}^a\,{(2-{m\over a})_{m-1} \over m!}zm=\sum_{m=1}^{\infty}\,b_m(a)z^m+\mathbf{O}(z^{a+1}),\;|z|<1.NEWLINE\]NEWLINE [\((x)_m=x(x+1)\cdots (x+m-1)\;(m\geq 1),\;(x)_0=1\) the standard Pochhammer symbols.]NEWLINENEWLINEIn particular NEWLINE\[NEWLINEb_1(a)=-a,\;b_2(a)={1-a\over 2},\;b_3(a)={(1-a)(2a-1)\over 6a}.NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThen for every \(\mu=1,2,\ldots,a-1\) there exists a positive constant \(c_{\mu}=c_{\mu}(a)\), such that for every non-negative integer \(n\) NEWLINE\[NEWLINE|p_{n,\mu}-q_n\alpha_{\mu}|\leq {c_{\mu}\over n^{a/2+1/(2a)}}\exp{\left(\sum_{m=1}^{a-1}\,(-1)^m b_m(a) \cos{({2\pi m\over a})} n^{1-m/a} \right)}.NEWLINE\]NEWLINENEWLINENEWLINEMoreover, \(D_n^{\mu}\cdot p_{n,\mu}\in\mathbf{Z}\), where \(D_n=\mathbf{LCM}(1,2,\ldots,n)\) [the least common multiple], and the following asymptotic formula holds: NEWLINE\[NEWLINEq_n={n!\over \sqrt{a}(2\pi )^{(a-1)/2}n^{a/2+1/(2a)}}\exp{\left(\sum_{m=1}^a\,(-1)^mb_m(a)n^{1-m/a} \right)} (1+\mathbf{O}(n^{-1/a})),\;n\rightarrow\infty.NEWLINE\]NEWLINENEWLINENEWLINE\vskip0.2cm The sequences \(\{p_{n,\mu}/q_n\}_{n\geq 0}\) provide for \(\mu=1,2,\ldots,a-1\) good simultaneous rational approximations converging subexponentially to the numbers \(\alpha_{\mu}\) as can be seen from:NEWLINENEWLINE\textbf{Corollary 1.2} Let \(a\geq 2\) be an integer, then for \(\mu=1,2,\ldots,a-1\): NEWLINE\[NEWLINE\begin{multlined}\left|\alpha_{\mu}-{p_{n,\mu}\over q_n}\right|\leq c_{\mu} \exp{\left(\sum_{m=1}^{a-1}\,(-1)^m b_m(a)(\cos{({2\pi m\over a})}-1)n^{1-m/a}\right)}\\ <\exp{\left[a(\cos{({2\pi \over a})}-1)n^{1-1/a}(1+\mathbf{o}(1))\right]},\end{multlined}NEWLINE\]NEWLINE for \(n\rightarrow\infty\), where \(c_{\mu}=c_{\mu}(a)\) is a positive constant independent of \(n\).NEWLINENEWLINE\vskip0.2cm \textbf{Corollary 1.3} Let \(a\geq 2\) be an integer, let \(q_n\) be defined as before and let NEWLINE\[NEWLINEp_n=\sum_{k=0}^n\,{n\choose k}^a k!(aH_{n-k}-(a+1)H_k),\;n\geq 0.NEWLINE\]NEWLINE Then NEWLINE\[NEWLINE\left|\gamma-{p_n\over q_n}\right|<\exp{\left[a(\cos{({2\pi\over a})}-1)n^{1-1/a}(1+\mathbf{o}(1))\right]},\;n\rightarrow\infty.NEWLINE\]NEWLINENEWLINENEWLINE\vskip0.2cm After this introduction (also containing some properties of Bell polynomials and as examples the cases \(a=3\) and \(a=4\)), the layout of the paper is as follows:NEWLINENEWLINE\S2: Analytical constructionNEWLINENEWLINE\S3: Bernoulli polynomials [introducing a.o. intermediary integrals \(I_{n,\mu}(u)\) in terms of a Meijer \(G\)-function]NEWLINENEWLINE\S4: Properties of the integrals \(I_{n,\mu}(u)\)NEWLINENEWLINE\S5: Asymptotics of the integrals \(I_{n,a-1}(u)\)NEWLINENEWLINE\S6: Proof of Theorem 1.1NEWLINENEWLINEReferences [11 items]NEWLINENEWLINE\vskip0.2cm Although the new results do not yet prove the irrationality of \(\gamma\), they indicate that it is possible to improve upon known orders and speed of approximation.