There might be some bugs (which may be fixed or not in other Maple versions) - it is hard to tell without seeing an example. It would be easier if you just posted some code that could be tried, and expected results.

Alec

It is more likely you have typos in you code or translated the formula
not in a correct way - however can not say something on that (do not
work with that functions and do not have the article).

But looking into your example: the outer loop just produces different
inputs r ( 0 < r < 1/4) for the inner loop.

The inner loop leads to products of Gamma and powers of 2F1. Even if
some cancellation error is possible (may be healed by high precision)
I doubt that Maple makes an error in that case.


PS: it is better to work with the classical interface for such kind of
problems, there are less traps for false input (say multiplication is
missing or similar).

Edited: you inner (with the given parameters) loop produces
54*(r-2)^4*(r^2-2*r+2)^4*GAMMA(5+4*r)/(r-1)^16/GAMMA(r+5)^4

Running the inner loop with uppercase Hypergeom, an unsigned r and CS=L*C one gets
a result, which lets one guess the original formula used here (by varying L):

hypergeom([1, 1+L*C],[1+C],r)^L *
  GAMMA(1+(C+r)*L)/(GAMMA(r+1+L*C)^(L-1))/GAMMA((C+r)*L+1-3*r)*GAMMA(1+L*C)^(L-1);

Substituting postive integers for L and C this turns out to be a rational
function in r, up to factors involving the Gamma function:

  subs(L=4, C = 1, %);
  convert(%,StandardFunctions):
  factor(%):
  f:=unapply(%,r); 

                                            4   2           4
                   54 GAMMA(5 + 4 r) (r - 2)  (r  - 2 r + 2)
         f := r -> ------------------------------------------
                                   16             4
                            (r - 1)   GAMMA(r + 5)


The outer loop only defines test values, which may be sampled into an Array and
applies f to that. Or one does it follows:

  data:=[a = 1, C = 1, L = 4];

  for rangedb from -5 by 5 to 40 do 
    y := 10^((1/10)*rangedb); 
    R := C/y; 
    r := R/(a+L*R);
    eval(f(r),data);
  print('f'(r)=evalf[20](%)); #print('f'(eval(r,data))=evalf[20](%));
  end do:

Whether those are results coinciding with the article now is on you, since that
paper is not public available :-)

For your series problem: using series for hypergeometrics (if even allowed to
be used) becomes quite a mess for inputs x where abs(x) > 2/3 and Maple uses
other ways. However usually you do not need to care how Maple does it, just
call the function.

1st: avoid floats like C=1.000 if you have no real reason, try exact values.

2nd: try do use functions, it gives better structure

3rd: if having doubts and using test values, then try simple ones or just one
(which can be typed in without running a script).

4th: avoid 'displayprecision' if there is no need for it, try evalf[3]().

Hm ... and using Array (upper case) is more concurrent ...


Now again:

  a := 1: C := 1: L := 4: CS := C*L: # avoid floats ...
  
  # your formula:
  Product(
    GAMMA(1+CS)*GAMMA(1+CS+(L-n+1)*r)*Hypergeom([1+CS, 1], [1+C], r)/
      (GAMMA(r+1+CS)*GAMMA(1+CS+(L-n)*r)), n = 1 .. L-1)*
    Hypergeom([1+CS, 1], [1+C], r);  
  #simplify(%); # not really needed, can be done later ...
  value(%);
  convert(%,StandardFunctions); 
  factor(%);
  f:=unapply(%,r);

                             4   2           4
                   54 (r - 2)  (r  - 2 r + 2)  GAMMA(5 + 4 r)
         f := r -> ------------------------------------------
                                    16             4
                            (-1 + r)   GAMMA(r + 5)


Your values:

  theR:=Array(1 .. 10):
  
  ind := 1:
  for rangedb from -5 by 5 to 40 do 
    y := 10^((1/10)*rangedb); 
    R := C/y; 
    r := R/(a+L*R); #r:='r':
    theR[ind] := r; 
    ind := ind+1 
  end do:
  r:='r':

Do not forget to reset your variables ... I would do it another way.

  theR:=evala(theR);

This gives the exact values for your various r. For all of them you
claim that different results are to be given. So take one, the 2nd
is quite simple, it is r = 1/5.

  f(1/5);        # strange, what Maple does here? No, due to Gamma.
  simplify(%);
  evalf[200](%): # use high precision in case of doubts
  evalf(%);      # cut down do working precision

                             11.07390057

Thx for the paper, however I do not see your formula and the notations
in the article are quite technical, and I do not want to step into the
details there. Well, I do not even see, which coefficients and inputs
you have to use F_A in eq (1) of the paper.

However I am sure your code is not correct, i.e. Product()* 2F1 is not
what you want.


For the convergence problem (approaching abs = 1): this already occures
if working with hypergeometric series, thus Maples does not use it in
a direct, simple way.