A condition for when $(a_n+b_n\Delta)(c_n+d_n\Delta)y_n = (c_n+d_n\Delta)(a_n+b_n\Delta)y_n$.

Question

The biconditional is $(a_n+b_n\Delta)(c_n+d_n\Delta)y_n = (c_n+d_n\Delta)(a_n+b_n\Delta)y_n$ iff $d_n/b_n = \Delta c_n / \Delta a_n = A$, where $A$ is a constant.

I showed the if part, but I am finding it difficult to prove the only if part; I got so far to the following:

$$(a_nd_n - c_nb_n)\Delta y_n = -b_n[c_{n+1}\Delta y_n +y_n \Delta c_n + d_{n+1}\Delta\Delta y_n + (\Delta y_n)\Delta d_n]+d_n[a_{n+1}\Delta y_n +(\Delta a_n)y_n+b_{n+1}\Delta\Delta y_n +(\Delta b_n)\Delta y_n$$

how from the last identity can I show that necessarily there exists a constant $A$ s.t. $d_n/b_n = \Delta c_n / \Delta a_n = A$?

Thanks.

Edit: just in case you don't understnad: $\Delta y_n = y_{n+1} - y_n$.

Answer 1

Here is a slightly different approach which might be helpful. We start with a few notational conventions which help to simplify calculations.

We consider $a,b,c$ and $d$ as functions with domain $\mathbb{N}$ with e.g. $a(n):= a_n$ for $n\in\mathbb{N}$. It is also convienient to consider an expression $(a+b\Delta)$ as operator acting on a function $y$ via \begin{align*} (a+b\Delta)y(n)&=a(n)y(n)+b(n)\left(y(n+1)-y(n)\right)\\ &=a_ny_n+b_n(y_{n+1}-y_n) \end{align*} We use the shift operator $\operatorname{E}$, defined as $\operatorname{E}y(n):=y(n+1)$ which is connected with the $\Delta$-operator via \begin{align*} \Delta&=\operatorname{E}-1\\ \Delta y(n)&=y(n+1)-y(n)\\ &=y_{n+1}-y_{n}\\ (\operatorname{E}-1) y(n)&=\operatorname{E}y(n)-y(n)\\ &=y(n+1)-y(n) \end{align*}

Now we are ready to go.

We have to find conditions among which the operator \begin{align*} a+b\Delta\qquad\text{and}\qquad c+d\Delta \end{align*} commute. We obtain

\begin{align*} (a+b\Delta)(c+d\Delta)&=(c+d\Delta)(a+b\Delta)\\ (a+b(\operatorname{E}-1))(c+d(\operatorname{E}-1))&=(c+d(\operatorname{E}-1))(a+b(\operatorname{E}-1))\tag{1}\\ (a-b+b\operatorname{E})(c-d+d\operatorname{E})&=(c-d+d\operatorname{E})(a-b+b\operatorname{E})\tag{2}\\ (a-b)d\operatorname{E}+b\operatorname{E}(c-d)+b\operatorname{E}d\operatorname{E} &=d\operatorname{E}(a-b)+(c-d)b\operatorname{E}+d\operatorname{E}b\operatorname{E}\tag{3}\\ \end{align*}

Comment:

• In (1) we use $\Delta=\operatorname{E}-1$.

• In (2) we observe $(a-b)(c-d)$ cancels out.

Now we apply both sides of (3) to the function $y$. We obtain \begin{align*} &\left((a-b)d\operatorname{E}\right)y(n)+\left(b\operatorname{E}(c-d)\right)y(n) +\left(b\operatorname{E}d\operatorname{E}\right)y(n)\\ &\qquad=\left(d\operatorname{E}(a-b)\right)y(n)+\left((c-d)b\operatorname{E}\right)y(n) +\left(d\operatorname{E}b\operatorname{E}\right)y(n)\\ &(a_n-b_n)d_ny_{n+1}+b_n(c_{n+1}-d_{n+1})y_{n+1}+b_nd_{n+1}y_{n+2}\\ &\qquad=d_n(a_{n+1}-b_{n+1})y_{n+1}+(c_n-d_n)b_ny_{n+1}+d_nb_{n+1}y_{n+2}\tag{4} \end{align*}

We observe from (4) the operators commute if and only if the coefficient functions of $y_{n+1}$ and $y_{n+2}$ are equal on both sides. Comparing the coefficients of $y_{n+2}$ we see \begin{align*} b_nd_{n+1}=d_nb_{n+1}\\ \end{align*} from which condition $\frac{d_n}{b_n}=\frac{d_{n+1}}{b_{n+1}}=\text{const.}$ follows.

Using this relationship when comparing the coefficients of $y_{n+1}$ in (4) we obtain \begin{align*} (a_n-b_n)d_n+b_n(c_{n+1}-d_{n+1})&=d_n(a_{n+1}-b_{n+1})+(c_n-d_n)b_n\\ a_nd_n+b_nc_{n+1}&=d_na_{n+1}+c_nb_n\\ b_n(c_{n+1}-c_n)&=d_n(a_{n+1}-a_n)\\ b_n\Delta c_n&=d_n\Delta a_n \end{align*}

and the claim \begin{align*} \frac{d_n}{b_n}=\frac{\Delta c_n}{\Delta a_n}=\text{const.} \end{align*} follows.