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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.

##### MSC:
 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.)
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##### References:
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