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Rational approximations to values of Bell polynomials at points involving Euler’s constant and zeta values. (English) Zbl 1251.11050
Generalizing the construction from their paper [On a continued fraction expansion for Euler’s constant (Preprint, 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 \[ \left(\gamma,1!(2a+1)\zeta(2),2!\zeta(3),\ldots,(m-1)!(a+(-1)^ma)\zeta(m)\right) \] for \(m=1,2,\ldots,a\) and \(a\in\mathbf{N}\).

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 \[ \exp{\left(\sum_{m=1}^{\infty}\,x_m{t m\over m!}\right)}=\sum_{n=0}^{\infty}\,Y_n(x_1,\ldots,x_n){tn\over n!}, \] with explicit representation \[ Y_n(x_1,\ldots,x_n)=\sum_{\pi(n)}\,{n!\over k_1!\cdots k_n!}\left({x_1\over 1!}\right)^{k_1}\cdots\left({x_n\over 1!}\right)^{k_n}, \] where the summation is taken over all partitions \(\pi(n)\) of \(n\) into \(n\) non-negative integers \(k_j\) with \[ \sum_{j=1}^n\,jk_j=n. \]
The main results are now:
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 \[ q_n:=\sum_{k=0}^n\,{n\choose k}^ak!\in\mathbf{Z},\;p_{n,\mu}:=\sum_{k=0}^n\,{n\choose k}^ak! Y_{\mu}(r_1(k),\ldots, r_{\mu}(k))\in\text\textbf{Q}, \] where \[ r_m(k)=(m-1)! (aH_{n-k}^{(m)}+(-1)^mH_k^{(m)}),\;k=0,1,\ldots,n. \] [\(H_n^{(m)}=\sum_{k=1}^n\,1/k^m,\;m\geq 1; H_0=0.\)]
Let \[ \alpha_{\mu}:=Y_{\mu}\left(\gamma,1!(2a-1)\zeta(2),2!\zeta(3),\ldots,(\mu-1)!(a+(-1)^{\mu}(a-1))\zeta(\mu)\right). \]
Suppose that the coefficients \(b_m(a)\) are defined by \[ -a\log{\left(1+\sum_{m=1}^a\,{(2-{m+1\over a})_m\over (m+1)!}z^m\right)}-\sum_{m=1}^a\,{(2-{m\over a})_{m-1} \over m!}zm=\sum_{m=1}^{\infty}\,b_m(a)z^m+\mathbf{O}(z^{a+1}),\;|z|<1. \] [\((x)_m=x(x+1)\cdots (x+m-1)\;(m\geq 1),\;(x)_0=1\) the standard Pochhammer symbols.]
In particular \[ b_1(a)=-a,\;b_2(a)={1-a\over 2},\;b_3(a)={(1-a)(2a-1)\over 6a}. \]

Then 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 \(n\) \[ |p_{n,\mu}-q_n\alpha_{\mu}|\leq {c_{\mu}\over n^{a/2+1/(2a)}}\exp{\left(\sum_{m=1}^{a-1}\,(-1)^m b_m(a) \cos{({2\pi m\over a})} n^{1-m/a} \right)}. \]
Moreover, \(D_n^{\mu}\cdot p_{n,\mu}\in\mathbf{Z}\), where \(D_n=\mathbf{LCM}(1,2,\ldots,n)\) [the least common multiple], and the following asymptotic formula holds: \[ q_n={n!\over \sqrt{a}(2\pi )^{(a-1)/2}n^{a/2+1/(2a)}}\exp{\left(\sum_{m=1}^a\,(-1)^mb_m(a)n^{1-m/a} \right)} (1+\mathbf{O}(n^{-1/a})),\;n\rightarrow\infty. \]

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:
Corollary 1.2 Let \(a\geq 2\) be an integer, then for \(\mu=1,2,\ldots,a-1\): \[ \begin{split}\left|\alpha_{\mu}-{p_{n,\mu}\over q_n}\right|\leq c_{\mu} \exp{\left(\sum_{m=1}^{a-1}\,(-1)^m b_m(a)(\cos{({2\pi m\over a})}-1)n^{1-m/a}\right)}\\ <\exp{\left[a(\cos{({2\pi \over a})}-1)n^{1-1/a}(1+\mathbf{o}(1))\right]},\end{split} \] for \(n\rightarrow\infty\), where \(c_{\mu}=c_{\mu}(a)\) is a positive constant independent of \(n\).

Corollary 1.3 Let \(a\geq 2\) be an integer, let \(q_n\) be defined as before and let \[ p_n=\sum_{k=0}^n\,{n\choose k}^a k!(aH_{n-k}-(a+1)H_k),\;n\geq 0. \] Then \[ \left|\gamma-{p_n\over q_n}\right|<\exp{\left[a(\cos{({2\pi\over a})}-1)n^{1-1/a}(1+\mathbf{o}(1))\right]},\;n\rightarrow\infty. \]

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:
§2: Analytical construction
§3: Bernoulli polynomials [introducing a.o. intermediary integrals \(I_{n,\mu}(u)\) in terms of a Meijer \(G\)-function]
§4: Properties of the integrals \(I_{n,\mu}(u)\)
§5: Asymptotics of the integrals \(I_{n,a-1}(u)\)
§6: Proof of Theorem 1.1
References [11 items]

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.

11J13 Simultaneous homogeneous approximation, linear forms
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
11B68 Bernoulli and Euler numbers and polynomials
11M35 Hurwitz and Lerch zeta functions
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
Full Text: DOI
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