How to work properly with assumptions?

Question

I'm trying to find an analytical solution which depends only of $i,j$ and $a$ variables, for a serie considering some initial assumptions.

The series is

$$s_{i0}=s_{0j} = 1,\\ s_{11} = 1-a,\\ s_{12} = a,\\ s_{ij} = \sum_{k=1}^2 s_{1k}s_{(j-k)i},$$

for $i = 2\ldots N$, $j=1\ldots 2$, $s_{i0}=s_{0j} = 1$ and $s_{ij}=0$ if $i<0$ or $j<0$.

I tried coding like this:

s[1, 1] := 1 - a;
s[1, 2] := a;
s[i, 0] := 1;
s[0, j] := 1;

s[i, j] = Refine[Sum[s[1, k] s[i - k, j], {k, 1, 2}],Assumptions -> {i >= 2, 1 <= j <= 2}]

But I only get the result

a s[-2 + i, j] + (1 - a) s[-1 + i, j]

which is in function of the previous $s_{ij}$ and its look like the assumptions are not taken in account (because of the s[-2 + i, j] and s[-1 + i, j] which are assuming negative index values).

I'm new in Mathematica, and if you guys could help I'll be much appreciated.

Answer 1

Although it is okay to use the ":=" or SetDelayed command, the plain Set command "=" may be simpler for the first few statements, because they have constants on the RHS. So, we begin with

ClearAll["Global`*"]
s[1, 1] = 1 - a;
s[1, 2] = a;

For the next two statements we need to use the Blank[] expression, which is abbreviated with the underscore, as

s[_, 0] = 1;
s[0, _] = 1;

In the above, there is no need to name the blank, since the name does not appear on the RHS.

Now we can code

s[i_, j_] := Simplify[Sum[s[1, k] s[i - k, j], {k, 1, 2}]]

Here, we need to name the blanks and use the the ":=" symbol, because $i$ and $j$ appear on the RHS. Also, we could have coded this SetDelayed before the Set statements, but the more general cases are usually defined after the more specific cases.

Now we can try to find s[1,3], but if we do we will get an error, because we have left something out. When $i$ is 1 and $k$ is 2, we get an s[-1,3] in our sum, and that is a problem.

So, we can add the statement s[-1,_] = 0;
Now we can make a table of values of $s_{ij}$ like this

t = Table[s[irow, jcol], {irow, 0, 4}, {jcol, 1, 3}]
t // TableForm

$\begin{array}{ccc} 1 & 1 & 1 \\ 1-a & a & 1-a \\ a^2-a+1 & -(a-2) a & a^2-a+1 \\ -a^3+a^2-a+1 & a \left(a^2-2 a+2\right) & -a^3+a^2-a+1 \\ a^4-a^3+a^2-a+1 & -a \left(a^3-2 a^2+2 a-2\right) & a^4-a^3+a^2-a+1 \\ \end{array}$

Edit 1:

Also, since the expression for s[i,j] is a recursion relation, to avoid recalculating the values of $s_{ij}$ you may want to use

s[i_, j_] := s[i, j] = Sum[s[1, k] s[i - k, j], {k, 1, 2}] // Simplify

instead of the first definition. This new definition tells MMA to save the intermediate values, which uses more memory, but will be faster if you need to calculate many values $s_{ij}$ for large $i$ and $j$.

Edit 2:

Now claim there are only two unique columns in any table t with any number of rows and columns calculated as above. This conjecture seems obvious by inspection using a large number of rows and column. Here's how we can verify the claim for a few rows and columns. First take the transpose, which gives a list of the columns. Then drop the second element and use Union to get a list of unique columns. There is only one element, so there is only one unique column, except for the second column. Here's the code:

t = Table[s[irow, jcol], {irow, 0, 50}, {jcol, 1, 30}];
Drop[Transpose[t], 2] // Union // Length

A second conjecture is the $i$-th row of the first column can be written as $s_{i,1}=\Sigma_{k=0}^i(-a)^k$. We can find another form for this expression with

Sum[(-a)^k, {k, 0, i}]//Factor

$\frac{a (-a)^i+1}{a+1}$

That's the closed form for the $s_{i,j\neq 2}$. We can get a closed form for $s_{i2}$ by a third conjecture that the $i$-th row of the second column of table $t$ is $s_{i,2}=2+(-a)^i-2\Sigma_{k=0}^i(-a)^k$. Here's the code

Factor[ 2 + (-a)^i - 2 Sum[(-a)^k, {k, 0, i}] ]

$\frac{(-a)^i+(-a)^{i+1}+2 a}{a+1}$

To prove the conjectures using Mathematica, searching this forum for "induction" would be a good first step. Also, replacing $a$ with $-b$ from the beginning could make the expressions simpler in an induction proof.