Recursive definition of nested functions

Question

This is probably very easy, but I cannot figure out a way to define a function like: $$ g_h = ( 1 + f_1(1+f_2(1+f_3(.. (1 + f_h)))))$$

@Mr. Wizard, if you put the word "fold" in somewhere, sure… otherwise no. ;) — J. M.'s missing motivation, Mar 08 '16 at 15:41

score 7 · Accepted Answer · answered Mar 08 '16 at 15:39

7

Fold should work for you:

Fold[1 + #2[#1] &, x, Reverse @ {f1, f2, f3, f4}]

1 + f1[1 + f2[1 + f3[1 + f4[x]]]]

answered Mar 08 '16 at 15:39

Mr.Wizard

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I was about to submit this when your answer showed up. This gives OP's more exactly I think: g[h_] := Fold[1 + Subscript[f, #2][#1] &, 1 + Subscript[f, h], Reverse@Range@(h - 1)] – Jason B. Mar 08 '16 at 15:41
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@JasonB If the OP wants to generate this expression for typesetting that will help. If writing a program I would recommend avoiding subscripts. – Mr.Wizard Mar 08 '16 at 15:42
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People like their subscripts, who are we to fight? :-P – Jason B. Mar 08 '16 at 15:46
@Mr-Wizard, can you point me to a post on here where it clearly shows the pitfalls of using Subscript indiscriminately, in a common situation? I don't use it myself, but it was years before I figured out that f[1] was the right substitute for $f_1$. Until then, I was doing ToExpression["f"<>IntegerString[1]] in my loops, which is cumbersome. – Jason B. Mar 08 '16 at 15:53
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@JasonB point #3 in (18395) is the first thing that comes to mind. Perhaps there is more. I simply know that many experienced users have cautioned against this, and early in the process of learning Mathematica I had a number of problems which I later traced to the use of subscripts. As a result I reserve Subscript almost entirely for formatting. – Mr.Wizard Mar 08 '16 at 15:56

score 3 · Answer 2 · answered Mar 08 '16 at 15:50

So the way to get exactly what is written is

g[h_] := Fold[1 + Subscript[f, #2][#1] &, 1 + Subscript[f, h], Reverse@Range@(h - 1)]

so that

g[5]

gives

But as is pointed out all the time here, you should avoid subscripts. Anytime you want to use $f_i$, you should use f[i] instead. So in this case what you need is

g[h_] := 
 Fold[1 + f[#2][#1] &, 1 + f[h], Reverse@Range@(h - 1)]
g[5]
(* 1 + f[1][1 + f[2][1 + f[3][1 + f[4][1 + f[5]]]]] *)

Less readable, but now you don't have to worry about what a DownValue of Subscript is.

Recursive definition of nested functions

2 Answers2