First order linear recurrence relation

I would like to solve the following first order non-homogenous recurrence relation $$ \left(n^2+n-1\right) y_{n+1}-(n-2)\,n\,y_n=\frac{1}{256} \left(25 n^3+150 n^2-173 n+32\right)\,. $$

It is a simple exercise to solve the associated homogenous recurrence relation $$ \left(n^2+n-1\right)\,y^H_{n+1}-(n-2)\,n\,y^H_n=0\,, $$ which yields $$ y^H_n = C_1\,\frac{\Gamma (n-2) \Gamma (n)}{\Gamma \left(n-\frac{\sqrt{5}}{2}+\frac{1}{2}\right) \Gamma \left(n+\frac{\sqrt{5}}{2}+\frac{1}{2}\right)}\,, $$ where $C_1$ is a constant and $\Gamma$ is a gamma function.

I tried to use telescoping to solve for $y_n$. In order to do this I set $$ y_n = y_n^H+y_n^H z_n $$ which gives for $z_n$ $$ z_{n}-z_{n-1}=\frac{\left(25 n^3+75 n^2-398 n+330\right)}{256} \frac{\Gamma \left(n-\frac{\sqrt{5}}{2}-\frac{1}{2}\right) \Gamma \left(n+\frac{\sqrt{5}}{2}-\frac{1}{2}\right)}{\Gamma (n-2) \Gamma (n)}\,. $$ However, I was not able to find a closed form for $$ \sum_{k=1}^n \frac{\left(25 k^3+75 k^2-398 k+330\right)}{256} \frac{\Gamma \left(k-\frac{\sqrt{5}}{2}-\frac{1}{2}\right) \Gamma \left(k+\frac{\sqrt{5}}{2}-\frac{1}{2}\right)}{\Gamma (k-2) \Gamma (k)}. $$

Any other ideas?

Solution 1:

Let $\;y_n=u_n+v_n,\;$ where $$v_n=\dfrac{5n^2+35n-56}{256},\quad (n^2+n-1)v_{n+1}-(n-2)nv_n = \dfrac{25n^2+150n^2-173n+16}{256}.\tag1$$ Then $$(n^2+n-1)u_{n+1}-(n-2)nu_n = \dfrac1{16}.\tag2$$

As it was shown in the OP, solution of the homogeneous recurrence relation is $$u^\,_H=\dfrac{\Gamma(n)\Gamma(n-2)}{\Gamma(n+\varphi)\Gamma(n-\varphi+1)},\tag3$$ where $\;\varphi=\dfrac{\sqrt5+1}2\;$ is the golden ratio.

Substitution $\;u=u^\,_H w\;$ leads to the equation $$w_{n+1}-w_n = \dfrac{\Gamma(n+\varphi)\Gamma(n-\varphi+1)}{16\Gamma(n+1)\Gamma(n-1)}.\tag4$$

Therefore, $$y_n=C\,\dfrac{\Gamma(n)\Gamma(n-2)}{\Gamma(n+\varphi)\Gamma(n-\varphi+1)}+\dfrac{5n^2+35n-56}{256}+\sum\limits_{k=m}^{n-1} \dfrac{\Gamma(k+\varphi)\Gamma(k-\varphi+1)}{16\Gamma(k+1)\Gamma(k-1)},\tag5$$ where the parameters $\;C,m\;$ should be defined from addition conditions.

Presentation $(5)$ is not a closed form, but it looks as the maximum of possible.