The statement of the problem:
In the following formula,
$$g(u,v) - \sum_{\Delta,l} c_{\Delta,l} u^{\frac{1}{2}} G^{(l)}\Bigg(\frac{1}{2} (\Delta-l),\frac{1}{2} (\Delta-l),\Delta,u,v \Bigg) = 0$$
what we want to calculate are the values of the constant $c_{\Delta,l}$ when we Taylor expand $g(u,v)$ and we know everything else.
It convenient to change basis from $\Delta,l$ to $m,t$ using the relations $l=2m$ and $\Delta-l=2(t+1)$. The reason for doing so is that $m$ goes from zero to infinity taking all integer values, and $t$ starts from one to infinity taking all integer values.
The code: These are just definitions
aa = 1/2 (\[CapitalDelta] - l);
l = 2 m;
\[CapitalDelta] = 2 (t + m + 1);
u = x z;
v = (1 - x) (1 - z);
g[u_, v_] := u + u/v
This is the definition of $u^{\frac{1}{2}} G^{(l)}$
ConfBlock[aa_, aa_, \[CapitalDelta]_, x_, z_] := (x z)^(
1/2 (\[CapitalDelta] - l))/(
z - x) ((-(1/2) z)^
l z Hypergeometric2F1[1/2 (\[CapitalDelta] + l),
1/2 (\[CapitalDelta] + l), \[CapitalDelta] + l,
z] Hypergeometric2F1[1/2 (\[CapitalDelta] - l - 2),
1/2 (\[CapitalDelta] - l - 2), (\[CapitalDelta] - l - 2),
x] - (-(1/2) x)^
l x Hypergeometric2F1[1/2 (\[CapitalDelta] + l),
1/2 (\[CapitalDelta] + l), \[CapitalDelta] + l,
x] Hypergeometric2F1[1/2 (\[CapitalDelta] - l - 2),
1/2 (\[CapitalDelta] - l - 2), (\[CapitalDelta] - l - 2), z] )
And this is how I define the original formula that I wrote to get the coefficients.
coefficient =
CoefficientList[
Series[g[u, v], {x, 0, 5}, {z, 0, 5}] -
Sum[c[m, t] (ConfBlock[aa, aa, \[CapitalDelta], x, z]) //
Simplify, {m, 0, 4}, {t, 1, 4}], {x, z}]
This is the result of the above
I only used a limited set of values, and hence I can go by "hand" and calculate the constants; i.e
Solve[1 - c[0, 1] == 0, c[0, 1]]
{{c[0, 1] -> 1}}
Solve[(1 - 9/10 c[0, 1] - 1/4 c[1, 1] == 0) /. c[0, 1] -> 1, c[1, 1]]
{{c[1, 1] -> 2/5}}
and so on and so forth.
What I would like:
I would like to give a command to Mathematica, to solve for all the coefficients, so that I take the results, I create a list of them and try to see if there is a sequence giving these numbers.
My failed attemps at a solution:
Using Solve
In[15]:= Solve[coefficient == 0, c[m, t]]
Out[15]= {}
Using SolveAlways
In[16]:= SolveAlways[coefficient == 0, c[m, t]]
Out[16]= {}
Thank you in advance.
P.S: This is what I found most useful from the comments.
coefficient
is{0, 2, 1, 1, 1, 1}
, which has nonzero entries. The{}
you get, (and which you would get if you replaced0
incoefficient == 0
with a zero matrix of the appropriate dimensions), indicates there are no values of thec[i, j]
that make all the entries0
. $\endgroup$ – Michael E2 Jan 27 '18 at 13:17c[m, t]
in yourSolve[...]
command is treated as a literal expression and not a pattern to be matched. It should be, I think,Variables[coefficient]
. $\endgroup$ – Michael E2 Jan 27 '18 at 13:20Solve
that not all the variablesc[m,y]
has solution. Do you expect this? It seems possible for the example you have provided... $\endgroup$ – José Antonio Díaz Navas Jan 27 '18 at 13:58