This is solvable using callgraph data generated by Clang.
Step 1. Generate call graph information using clang: 
clang -S -emit-llvm SourceFile.c -o - | opt -analyze -print-callgraph 
(From Generate calling graph for C++ code, replacing -dot-callgraph with -print-callgraph.)
For an input like:
void a(){}
void b(){a();}
void c(){a(); b();}
void d(){a(); c();}
void e(){e();}
this will produce:
CallGraph Root is: <<null function: 0x0x7fdef25036c0>>
Call graph node <<null function>><<0x7fdef25036c0>>  #uses=0
  CS<0x0> calls function 'a'
  CS<0x0> calls function 'b'
  CS<0x0> calls function 'c'
  CS<0x0> calls function 'd'
Call graph node for function: 'a'<<0x7fdef2503750>>  #uses=4
Call graph node for function: 'b'<<0x7fdef25037d0>>  #uses=2
  CS<0x7fdef2500a38> calls function 'a'
Call graph node for function: 'c'<<0x7fdef2503870>>  #uses=2
  CS<0x7fdef2500cb8> calls function 'a'
  CS<0x7fdef2500d28> calls function 'b'
Call graph node for function: 'd'<<0x7fdef2503970>>  #uses=1
  CS<0x7fdef2500fe8> calls function 'a'
  CS<0x7fdef2501058> calls function 'c'
Call graph node for function: 'e'<<0x7f8912d03c10>>  #uses=2
  CS<0x7f8912d01318> calls function 'e'
(In C++, mangled function names can be cleaned up with c++filt; templates get ugly but are doable.) With that data, it's possible to sketch out how one detects recursion:
Step 2. Parse callgraph data into favorite scripting language to form representation of callgraph.
class Graph(object):
  _callees = []
  def add_callee(self, f):
    self._callees.append(f)
    # etc
Step 3. For each function, walk graph looking for calls to that function. Something kind of like this:
def walkGraph(node,f,stack):
  for callee in node._callees:
    if f == callee:
      print('Recursion!')
      dumpStack(stack,f)
    else:
      walkGraph(callee,f,stack.append(node))