Explain the concept of extended recursion and how it is used in programming.

Extended recursion is a technique where a recursive function not only calls itself but also uses additional mechanisms to optimize or extend the capabilities of recursion. For example, this can include memoization (caching call results), tail recursion, or combining recursion with iteration.

In PHP, extended recursion is often used to solve problems where simple recursion might lead to redundant calculations or stack overflow. For example, when calculating Fibonacci numbers with memoization:

function fib($n, &$memo = []) {
    if ($n <= 1) return $n;
    if (isset($memo[$n])) return $memo[$n];
    $memo[$n] = fib($n - 1, $memo) + fib($n - 2, $memo);
    return $memo[$n];
}

echo fib(10); // 55

Here, recursion is extended by using the $memo array to store already computed values, which significantly speeds up execution and reduces the load on the call stack.