I need to know the solution of this recurrence: $$2a[n,k-1]-a[n-1,k-1]-2a[n-1,k-2]+a[n-2,k-2]=a[n,k]$$
With the initial value unspecified.
Also, wolfram is good at solving linear 1D recurrence, but seems pretty lame on 2D version. Could you recommend me some 2D linear recurrence solver?
Could someone please help me? Thank you
$$a_{n,k}=2a_{n,k-1}-a_{n-1,k-1}-2a_{n-1,k-2}+a_{n-2,k-2}\tag{1}$$
Let us have a graphical understanding of the dependencies, in particular in order to have a clear idea about the initializing data.
Here, the "computability" of red box $a_{n,k}$ relies recursively on the computability of the yellow boxes $a_{n,k-1}$, etc. arranged as a "Tetrix-like" shape, with a South-East general "sweeping" direction.
As a consequence, if you place the yellow shape at any place on the West and North borders, the process will be effective iff you have given initial values to all green boxes (on two levels) (as noted by @Alexander Burstein).
This infinite number of degrees of freedom prevents from giving a general solution.
Nevertheless, when I have simulated the creation of the table of the $a_{n,k}$ for many different initial values, a surprizingly common behavior is observable : the fact that, on the eastern border, one has always a same type of alternating values as can be seen on the example below (for $n=10$) :
where in this case, the significant values $a_{n,10}$ alternate in the following approximate way :
$$2.25, -7.06, 8.26,-4.73, 1.30, -0.16$$
What can be the explanation ?
Edit : you have suggested to take $n+k=m$ in (1), giving :
$$a_m=2a_{m-1}-a_{m-2}-2a_{m-3}+a_{m-4}$$
This linear recurrence has a characteristic polynomial
$$r^4-2r^3+r^2+2r-1=0$$
with solutions
$$r_1 \approx 0.46899, \ r_2 \approx -0.88320, \ r_3 \approx 1.55377e^{0.217i \pi}, \ r_4 \approx 1.55377e^{-0.217i \pi}$$
Therefore, the explicit solution of (2) has the following form :
$$a_m\approx \underbrace{a(0.46899)^m+b(-0.88320)^m}_{\text{tending to } 0}+(1.55377)^m(c \cos(m 0.217 \pi)+d \sin(m 0.217 \pi)\tag{1}$$
(for certain constants $a,b,c,d$ relying on initial conditions, here assumed uniform, for example with all green borders set to $1$). explaining the long-term exponential trend in $(1.55377)^n$ coupled with a certain periodicity.