implement lambdas recursively

Solution 1 is pretty good but it leaves a vulnerability for error.
It depends on the binding to the name factorialValue for the recursive call.
If were compromised it falls apart.
Very contrived example:

Func<int, int> Factorial = null;
Factorial = x => (x <= 0) ? 1 : x * Factorial(x - 1);
Func<int, int> incorrectFactorial = Factorial;
Factorial = x => x;

int ans = incorrectFactorial(5);

The value of ans here is 20 because of the reassignment of the identifier Factorial.
The name binding the recursive method needs to be protected:

Func<int, int> Factorial = null;
Factorial = x => {
  Func<int, int> internalFactorial = null;
  internalFactorial = xi => (xi <= 0) ? 1 : xi * internalFactorial(xi - 1);
  return internalFactorial(x);
};

Func<int, int> otherFactorial = Factorial;
Factorial = x => x;

int ans = otherFactorial(5);

Here the value of ans is the correct 120 because the recursion identifier was internal to the scope of the lambda.
Another way to accomplish this would be to have a method that returns a Func<int, int>:

static Func<int, int> GetFactorialFunction()
{
  Func<int, int> Factorial = null;
  Factorial = x => (x <= 0) ? 1 : x * Factorial(x - 1);
  return Factorial;
}

and invoke the returned function:

int ans = GetFactorialFunction()(5);

Of course, at this point using a lambda is probably superfluous since the function could be implemented directly as a named, recursive function.