@Pascal4QM 

Why did you use the commuator? It seems that anyway the formula must be proved by hand (via induction it's easy).

As it seems that you work with this package, and knowing that several Maple basic operators are redefined (`*` included), how can the user know whether a command outside the package (such as solve)  is reliable in the new context? Are these situations documented, or it's only "common sense"?

@alecjacobson 

The computations are simple; they can be easily included in a procedure:

Int( F(x,y[0],y[1],`...`,y[n] ), x=a..b);   # --> min, 

y[k] = Diff(y[0],[x$k]);                         # y[0] is the unknown function

y[k](a)=A[k], y[k](b)=B[k], k=0..n-1;   # boundary conditions

Sum( (-1)^k * Diff( Diff(F,y[k]), [x$k]), k=0..n ) = 0;
# E-L equation; order is 2*n, equal to the number of boundary conditions

@Adam Ledger 

I have explained that it's not possible. Try:

dsolve( diff(y(x),x) = x*y(x) + exp(y(x)), y(x) );

@taro 

As I said, the order cannot be monomial, and sort accepts only monomial orders. Maple will never display p in this form. Of course it is possible to construct a special sort routine and inert forms, but I don't thick it is worth doing it.

You must provide an order for the monomials, i.e. define when exactly x^A*y^B*z^C > x^a*y^b*z^c.

Note that your order cannot be a monomial order (in the sense of Groebner package).

@Carl Love 

Is there a strong reason to use &where instead of a simple comma?

Using a comma, eval(%) should substitute v[i], but strangely (for me), eval['recurse'] is needed.

Could you explain the evaluation mechanism for this?

In a simplified form:

restart;
t:=a+b,[b=c];
eval(t); # ==> a+b, [b = c]        (why?)
eval[recurse](t); # ==> a+c      (as expected).

@Adam Ledger 

Note that Whittaker's functions do not appear here; they are included only in the second edition of  "A Course of Modern Analysis" written with G.N. Watson.

@Abdoulaye 

You can define the intervals and the expressions (this can be done automatically in a procedure):

intervals:=-infinity..0, 0..1, 1..2, 2..3, 3..infinity:
expressions:= 0, t^2, 1, 3-t, 0:
max( seq( maximize(expressions[i],t=intervals[i]),i=1..nops([intervals])) );
    1

@Markiyan Hirnyk 

My point was that a person interested in Maple and minimal surfaces should be informed about the existence of this book (not very easy to find); probably I was completely wrong.

@Markiyan Hirnyk 

Your question is not answered in the book, but some other special surfaces are computed symbolically using Maple (V.5 and 6).

@Markiyan Hirnyk

It's the site of a Chinese university; I had no problems with it.

https://www.researchgate.net/publication/268347899_Constructing_Minimal_Triangular_Mesh_Based_on_Discrete_Mean_Curvature

or directly to the pdf:

http://www.math.zju.edu.cn/webpagenew/uploadfiles/attachfiles/2011112721225969.pdf

Edit.
Oprea J. - The mathematics of soap films_Explorations with Maple, AMS, 2000

@Adam Ledger 

Looking closer to your F, it is easy to see that actually F(s) = GAMMA(s), all the terms in the sum being 0 except for k=0.

If you are interested in such problems I'd recommend the book

Bressoud D., Wagon S. - A course in computational number theory, Wiley, 2000

It also contains Mathematica code but you could try to convert it into Maple.

@Adam Ledger 

@aamirkhan 

Yes, it seems that there exists an elementary integral wrt s. Your f_3 is a sum of >6000 terms, and each can be integrated simbolycally (but I did not verify all of them). Maple seems to refuse to integrate globally. You could use map(int,...)  but I don't think that the resulting HUGE expression will be useful.
Note also that using floats in symbolic computations is not usually a good idea.