This will do it:

> Q:= 1/2* LerchPhi(-1, 1, 1/2 - a/2) + 1/2* LerchPhi(-1, 1, a/2 + 1/2);
  simplify(convert(Q,hypergeom));

1/4*Pi/sin(1/4*Pi*a+1/4*Pi)/cos(1/4*Pi*a+1/4*Pi)

> combine(%);

1/2*1/cos(1/2*Pi*a)*Pi

Your code doesn't work at all: there is no such thing as AND (you mean and).  Also, I'm pretty sure you want {p,q}, not [p,q].  Note that

S := solve({X = .15*p*(1/2), L-X = .15*q*(1/2)}, {p, q})

returns (in the first iteration)

S := {p = 0., q = 13.33333333*L}

Then

p := S[1]; q:= S[2];

makes no sense at all: in this case it would assign p the equation p=0. (which is the source of your "too many levels of recursion".

Actually, if you might want to use p and q again as symbolic variables, you shouldn't assign them values at all.  You might try something like 

p1:= subs(S,p); q1:= subs(S,q);

and then use p1 and q1 instead of p and q in your if statement.

solve is unlikely to work for such complicated equations.  fsolve, to find a numerical solution, is the way to go.  I see two mistakes in your K3 right away: you're missing some multiplication signs.

t(51771823070780165/18014398509481984) is interpreted as a function t evaluated at 51771823070780165/18014398509481984,

That's what's likely causing the "invalid arguments" error.

3(1277474181801312151/22517998136852480000) is interpreted as the constant function 3 evaluated at 1277474181801312151/22517998136852480000.

The differential equation being singular at 0, both of the given boundary conditions may be problematic.  Another problem, maybe a bigger one, is the singularity of polylog(3/2,z) at z=1.  In order to avoid this (and have everything real) you need y(r) > 1194.3, which conflicts with the boundary condition y(infinity) = 0.  Or do you intend to have non-real solutions?

Maple has several functions for regression.  You might try Fit or LinearFit in the Statistics package, or LeastSquares in the CurveFitting package, or LeastSquares in the LinearAlgebra package.  For data in your form, the CurveFitting version is the simplest.  Try:

> CurveFitting(data, x);

You seem to be missing a "(" at the start of equation 1.

Your equations contain u only in the form of exp(b*(u-4300)) or exp(-b*(u-4300)), so I would suggest first making the substitution

u = ln(U)/b + 4300.  And then I would try fsolve rather than solve, to look for numerical solutions (it being very unlikely that solve would be able to find closed-form solutions).  However, fsolve couldn't find solutions either, and looking at the problem graphically it doesn't appear that there are any real solutions.  Do you have some reason to think that there are?

Actually seq produces an expression sequence.  Enclose it in square brackets to get a list. 

> L:= [seq(2*x+8,x=1..10)];

To convert a list to a table, you could use convert(..., table). 

> T:= convert(L, table);

Or is that not the kind of "table" you had in mind?

You can't do it with pointplot or pointplot3d, but you can use textplot or textplot3d.

Just to clarify acer's response: both in Maple 12 and 13 there is literally no such thing as GraphTheory[Trail], i.e. the GraphTheory package does not define an object with the name Trail.  The undefined function Trail is, however, used as a notation for the sequence of edges traversing a sequence of vertices in a given order, and this is understood by various GraphTheory commands.   In Maple 12, the Graph command accepts the Trail notation, but AddEdge does not (although it does in Maple 13).  You'd have to look at the help page for each individual command to see which does or doesn't.

It most likely means Maple can't find a solution.  If you post your equations we might be able to say more about the problem.

The constant 3.14... is Pi, not pi.  As for taking out the factor sigma^2, you could try factrix in my Maple Advisor Database

www.math.ubc.ca/~israel/advisor

1) How could three points not form a triangle?

2) Do you know how to find the distance between two points?

Yes, it is possible.  For example:
 

> f[0]:= 0: f[1]:= 1:
   for i from 2 to 30 do
      f[i]:= f[i-1] + f[i-2];
      if isprime(f[i]) then 
        print(cat(`#mn("`, f[i], `", mathcolor="red", fontweight="bold")`)) 
    else print(f[i]) fi
   od:

This should work in Standard; it won't work in Classic.

Write your DE as

> de:= diff(y(t),t$2) + y(t) = piecewise(t < Pi,0, t < 2*Pi, 1, 0);

and your initial conditions as

> ic:= y(0)=0, D(y)(0)=1;

Then see the help pages ?laplace and ?invlaplace.

You can use setoptions in the plots package.