# Summation equation

Recently encountered the following problem:

$$x_n = x_1 - 2 a \sum\limits_{i = 1}^{n-1} \left( 1 - \frac{a}{x_i} \right)$$,

where $$n>1$$, and $$x_1$$ is known. The task is to find $$x_n$$ for arbitrary n. It resembles an integral equation, but in a discrete form. What are the possible tricks to solve for that?

P.S.: Of course, one can try to solve it iteratively, but then the answer becomes cumbersome quickly. Although no one promised me that this particular problem has a nice and compact answer at all.

• I would start by evaluating $x_n-x_{n-1}$, it would give the nice recursive $x_n=x_{n-1}-2a-\frac{2a^2}{x_{n-1}}$. Commented Dec 15, 2022 at 0:30
• Also the transformation $y_n = x_n/a-1$ will reduce it to $y_n = y_{n-1}(y_{n-1}-1)/(y_{n-1}+1)$. Doubt that has an exact solution, but could do some interesting asymptotics (i.e. seems like generically $y_n \to 0$ but subgeometrically). Commented Dec 15, 2022 at 0:51
• Hi, @UdiFogiel I did that! And tried to look at that as a diff. equation, and it seems that it behaves somehow close to Lambert function, but need to have a closer look. Commented Dec 15, 2022 at 9:06

Thanks for the suggestions, most likely its impossible to solve it analytically in a discrete form, but in a continuous form - yes, you can make a good estimation.

First of all by considering $$y_{n+1}-y_{n}$$ we can get the following differential equation: $$y'(x)=-2a+\frac{2a^2}{y(x)}, \quad y(x=1)=x_1,$$ the solution of which is given by:

$$y(x) = a \left( W\left(\exp \left( -2 x + \frac{c_1}{a} -1 - \ln(a) \right) \right) - 1 \right),$$ where $$W(x)$$ is the Lambert W function, and the constant $$с_1$$ is found from the initial condition $$y(x=1)=x_1$$ (If anyone knows how to write it analytically - I would be grateful).

I compared a continuous solution with a discrete one, and in the limit I am interested in $$x_1 \gg a$$ it works for small $$n$$.

It is not that precise when $$x_1 \sim a$$, but very good at $$x_1 \gg a$$. Moreover, it seems that after that linearly decaying part, the continuous solution shows the average between $$x_n$$ for even $$n$$, and odd $$n$$. The only thing left is to estimate $$n$$ at which the linear decay stops.

• If I am not mistaken $$y(x)=a \Bigg[1+W\left(\frac{x_1-a }{a}\,\,e^{-2x+\left(\frac{x_1+a}{a}\right)}\right)\Bigg]$$ Nice problem and (+1). Commented Dec 15, 2022 at 14:26
• Long time no see :). Yet another problem from me with Lambert function. It comes from some quantum mechanical problem, and the fact that $x_n$ linearly decays with $n$ actually means that the model is bullshit, hehe. But we are working on modifying such that it works at least in some range of parameters. Commented Dec 15, 2022 at 19:58
• When I saw the problem, I thought that it could be related to QM. I am myself a quantum mechanicist (working more precisely with wave mechanics). For a year and half (long time ago) I have been LdB last student). I hope you see who he is (for me, my guru is still alive). Cheers :-) Commented Dec 16, 2022 at 2:42