On a constant occurred in a difference equation

In the talk, we consider the following difference equation which originates from a heating conduction problem. Let $\tau_1 = 1$ and
\begin{equation}\label{lab_1} \sum_{j=1}^{n}\left(\sum_{s=j}^{n}\tau_{s}\right)^{-1} = 1 \end{equation}
for $n\geqslant 2$. It is required to determine the asymptotic behaviour of $\{\tau_n \}$. This problem is an analog to the $1$-dimensional problem first studied by Myshkis [1]. We have a more general result for $\{\tau_n \}$ (see [2]).

Theorem 1. Assume $g: \mathbb{R^+} \to \mathbb{R^+}$ is a continuous and strictly decreasing function with $g(0^+) \ge 1$, and $ g(\infty)=0$. Let $\{\tau_n\}$ be a sequence of numbers recursively defined by
$$ \sum_{j=1}^{n}g\left(\sum_{s=j}^{n}\tau_{s}\right) = 1. $$
If $(\log g)''$ is nonnegative, then the sequence $\{\tau_n \}$ is increasing.

Theorem 1 asserts that $\{ \tau_n \}$ defined in (\ref{lab_1}) is increasing since $\log (x^{-1}) $ is convex in $\mathbb{R}^{+}$.
In 2013, we obtain [3] the following theorem.

Theorem 2. Let $\{\tau_n \}$ be the sequence recursively defined in (\ref{lab_1}). Then we have
$$ \tau_n = \log{n} + \gamma + O\biggl(\frac{1}{\log n}\biggr), $$
where $\gamma$ is the Euler's constant.

Since $ 4\pi au_0t_n= b \sum_{s=1}^n \tau_s, $ we immediately have the following:

Corollary 3. The heating-time sequence $\{t_n \}$ recursively defined by the heat equation (\ref{lab_1}) and the heating condition satisfies
$$ t_n= \frac{b}{4\pi a u_0} (n\log n +(\gamma -1)n) +O\biggl(\frac{n}{\log n}\biggr). $$


Applying the technique used in [3], we show that the following assertion holds true.

Theorem 4. Let $\{\tau_n \}$ be the sequence recursively defined in (\ref{lab_1}). Then we have
$$ \tau_{n} = \log{n} + \gamma + \frac{\delta}{\log{n}} + O\biggl(\frac{\log \log{n}}{(\log{n})^{2}}\biggr), $$
where
$$ \delta = \log 2+ \int_{1/2}^{1}\frac{1-x}{x^2} \log(1-x)\,dx + \sum_{j=2}^{\infty} \frac{(-1)^{j}}{j}\int_{0}^{1/2}\frac{x^{j-2}}{(1-x)^{j-1}}\,dx . $$
Here, $\delta < \infty $ since the series satisfies the criteria of convergence of alternating series.

[1] A.D. Myshkis, On a recurrently defined sequence, J. Difference Equ. Appl., 3 (1997), pp. 89–91.

[2] J.Y. Chen, On a difference equation motivated by a heat conduction problem, Taiwanese J. Math., 12 (2008), pp. 2001–2007.

[3] J.Y. Chen, Y. Chow, On the convergence rate of a recurrsively defined sequence, Math. Notes, 93 (2013) pp. 238–243.

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