When we look at how to use advanced searching methods for recursive functions, two options stand out: Ternary Search and Fibonacci Search. Each offers some advantages, but generally, Fibonacci Search is the better choice.

Performance and Efficiency
Fibonacci Search has a great way of searching called a logarithmic pattern. This means it works very efficiently with a time complexity of $O(\log n)$. Instead of just splitting the array in half, it uses Fibonacci numbers to break it into sections. This can result in fewer comparisons, which is super helpful for recursive functions that keep using this method.

On the other hand, Ternary Search also has a time complexity of $O(\log_3 n)$. But it splits the array into three parts, which can make it less efficient when using recursion due to more branches to manage.

Memory Usage
Both searching methods are designed to use memory well. However, Fibonacci Search needs less stack space during recursive calls. It doesn’t require keeping track of several branches like Ternary Search. Managing the stack is really important in recursive functions, so a method that uses fewer stack frames, like Fibonacci Search, is often a better option.

Implementation Complexity
Implementing Fibonacci Search can be a bit more complicated because you need to calculate Fibonacci numbers. But this complexity is usually worth it because of how well it works with recursion. Ternary Search is simpler to set up, but that doesn’t always make it the best choice for recursive tasks.

Conclusion
In the world of searching algorithms, both Ternary Search and Fibonacci Search can work with recursive functions. However, Fibonacci Search shines with its efficient time, lower memory usage, and compatibility with recursion. Still, the best method can depend on the specific problem and the data structure you’re working with. But when it comes to recursive functions, Fibonacci Search often proves to be the more effective choice.