Can recursion be replaced with a finite set of higher-order functions?
You may replace recursion with computation of fixed points, so all you need is a fixed-point operator

fix : (a -&gt; a) -&gt; a


which satisfies

fix f = f (fix f)


Then, given a recursive definition of f : a -&gt; b,

f = Φ(f)


we may define f as

f = fix (λg.Φ(g))


Here fix has type ((a -&gt; b) -&gt; (a -&gt; b)) -&gt; (a -&gt; b).

You may consider this to be cheating, but the question is not really research-level precisely because the connection between recursion and fixed points is a basic piece of knowledge. I am guessing that maybe you did not know about the connection.