Guessing Legendre Coefficients

SetOptions[Plot,AxesLabel->Automatic,PlotStyle->Thick,ImageSize->700,PlotRange->{-2,2},BaseStyle->{FontSize->16}];
SetOptions[Manipulate,BaseStyle->{FontSize->20}];
SetOptions[Manipulator,Appearance->{"Labeled"},BaseStyle->{FontSize->16}];
f[z__]=(1/8)-z-(33/4)*z^2+(105/8)*z^4;
SecretImage=;

The plotted function below (blue) is a linear combination of Legendre polynomials, (Subscript[P, l])(z), for different values of l (i.e. f(z)=(Subscript[c, 0])(Subscript[P, 0])(z)+(Subscript[c, 1])(Subscript[P, 1])(z)+(Subscript[c, 2])(Subscript[P, 2])(z)+(Subscript[c, 3])(Subscript[P, 3])(z)+(Subscript[c, 4])(Subscript[P, 4])(z)+...=(∑_( l =0)^∞)(Subscript[c, l])(Subscript[P, l])(z)):
 
Look at how the first few orders of Legendre polynomial (orange) compare to the given function:

Manipulate[
Grid[{
{Plot[{f[z],LegendreP[l,z]},{z,-1.5,1.5}]},
{},
{StringForm["(Subscript[P, ``])(z)=",l][ LegendreP[l,z]]}},Alignment->Left],
{l,0,10,1}]

1) What is the the order of a Legendre polynomial?

2a) Describe any patterns you see for odd and even Legendre polynomials.

2b) How does the number of “wiggles” in the plot of a Legendre polynomial correspond to the polynomial’s order?

2c) How does the order of a Legendre polynomial affect its end behavior?

3) Which orders do you expect to be present in the Legendre series for the given function?

Your job is to guess the values of the coefficients, (Subscript[c, l]).  To make your job (much) easier:
 - There are only 3 non-zero (Subscript[c, l]).
 - There are no terms of order greater than 4; l≤4. 
 - All non-zero (Subscript[c, l]) are positive or negative integers.
 - The magnitude of each (Subscript[c, l]) is no more than 4; |(Subscript[c, l])|≤4.

Manipulate[
myguess= (Subscript[c, 0]) LegendreP[0,z]+ (Subscript[c, 1]) LegendreP[1,z]+(Subscript[c, 2]) LegendreP[2,z]+(Subscript[c, 3]) LegendreP[3,z]+(Subscript[c, 4]) LegendreP[4,z];
plotmyguess=Plot[{"",myguess},{z,-1.5,1.5}];
plotf=Plot[f[z],{z,-1,1}];
myplot={plotmyguess,plotf};If[And[(Subscript[c, 0])==0,(Subscript[c, 1])==-1,(Subscript[c, 2])==2,(Subscript[c, 3])==0,(Subscript[c, 4])==3],myplot={plotmyguess,Graphics[Inset[SecretImage,{-1.5,-2},{0,0},{4,5}]],plotf}];
Grid[{{Grid[{
{StringForm["(Subscript[``P, 0])(z):",(Subscript[c, 0]) ],Plot[{"","",(Subscript[c, 0]) LegendreP[0,z]},{z,-1.5,1.5},ImageSize->200,Ticks->None]},
{StringForm["(Subscript[``P, 1])(z):",(Subscript[c, 1]) ],Plot[{"","",(Subscript[c, 1]) LegendreP[1,z]},{z,-1.5,1.5},ImageSize->200,Ticks->None]},
{StringForm["(Subscript[``P, 2])(z):",(Subscript[c, 2]) ],Plot[{"","",(Subscript[c, 2]) LegendreP[2,z]},{z,-1.5,1.5},ImageSize->200,Ticks->None]},
{StringForm["(Subscript[``P, 3])(z):",(Subscript[c, 3]) ],Plot[{"","",(Subscript[c, 3]) LegendreP[3,z]},{z,-1.5,1.5},ImageSize->200,Ticks->None]},
{StringForm["(Subscript[``P, 4])(z):",(Subscript[c, 4]) ],Plot[{"","",(Subscript[c, 4]) LegendreP[4,z]},{z,-1.5,1.5},ImageSize->200,Ticks->None]}}],
"  ",
Grid[{
{Show[myplot]},
{},
{"Legendre Series:"[ToString[(Subscript[c, 0])] "(Subscript[P, 0])(z)"+ToString[(Subscript[c, 1])]"(Subscript[P, 1])(z)"+ToString[(Subscript[c, 2]) ]"(Subscript[P, 2])(z)"+ToString[(Subscript[c, 3]) ]"(Subscript[P, 3])(z)"+ToString[(Subscript[c, 4]) ]"(Subscript[P, 4])(z)"]},
{},
{"Expanded Legendre Series:"[myguess]},
{},
{"Simplified Legendre Series:"[Simplify[myguess]]}},Alignment->Left]}}],
{{(Subscript[c, 0]),1},-4,4,1},{{(Subscript[c, 1]),1},-4,4,1},{{(Subscript[c, 2]),1},-4,4,1},{{(Subscript[c, 3]),1},-4,4,1},{{(Subscript[c, 4]),1},-4,4,1},ControlPlacement->Left]

4) How does the Legendre series compare to what you expected it to be?

By Corinne Manogue, Kerry Browne, Mike Vignal, and Ian Founds
Copyright 2000 Corinne Manogue