aa r X i v : . [ m a t h . G M ] F e b THE FA `A DI BRUNO FORMULA REVISITED
RAYMOND MORTINI
Abstract.
We present an intuitive approach to (a variant of) the Fa`a di Bruno formula whichshows how this formula may have been (re)discovered many times in history. Our representationfor the n -th derivative of the composition f ◦ g of two smooth functions f and g on R uses a simplersummation order so that the mysterious condition b +2 b + · · · + nb n = n in Fa`a di Bruno’s formuladoes not appear. How one may (re)discover oneself this formula
Let f and g be two functions on R whose n -th derivatives exist. The first 4 derivatives of f ◦ g are easy to calculate: ( f ◦ g ) ′ = ( f ′ ◦ g ) g ′ ( f ◦ g ) (2) = ( f (2) ◦ g )( g ′ ) + ( f ′ ◦ g ) g (2) ( f ◦ g ) (3) = ( f (3) ◦ g )( g ′ ) + 3( f (2) ◦ g ) g (2) g ′ + ( f ′ ◦ g ) g (3) ( f ◦ g ) (4) = ( f (4) ◦ g )( g ′ ) + 6( f (3) ◦ g ) g (2) ( g ′ ) + 4( f (2) ◦ g ) g (3) g ′ + 3( f (2) ◦ g )( g (2) ) + ( f ′ ◦ g ) g (4) The expressions get rapidly longer and more complicated. For example we have 42 summandsfor n = 10. Note that the number of terms equals the partition number p ( n ), that is the numberof ways (without order) of writing the integer n as a sum of strictly positive integers; by theHardy-Ramanujan formula we have p ( n ) ∼ n √ e π √ n/ .Let M j = { k = ( k , . . . , k j ) ∈ ( N ∗ ) j , k ≥ k ≥ · · · ≥ k j ≥ } be the set of ordered multi-indeces in N ∗ = { , , . . . } . If g is a function defined on R , and if g ( n ) isthe n -th derivative of g , then we denote by g ( k ) the function Q ji =1 g ( k i ) , where k = ( k , . . . , k j ) ∈ M j . Also, g (0) is, by convention, equal to the function g itself.The classical Fa`a di Bruno formula from ca. 1850 gives an explicit formula for ( f ◦ g ) ( n ) :( f ◦ g ) ( n ) ( x ) = X n ! b ! b ! · · · b n ! f ( j ) ( g ( x )) n Y i =1 g ( i ) ( x ) i ! ! b i , where the sum is taken over all different solutions in nonnegative integers b , b , . . . , b n of b + 2 b + · · · + nb n = n and j := b + · · · + b n .A nice historical survey on this appeared in [2]. See also [1].Without being aware of that formula, I developed around 1976-1980 the following formula:(1.1) ( f ◦ g ) ( n ) ( x ) = n X j =1 f ( j ) ( g ( x )) (cid:18) X k ∈ M j | k | = n C n k g ( k ) ( x ) (cid:19) , Mathematics Subject Classification.
Primary 26A24. where C n k = (cid:18) n k (cid:19)Y i N ( k , i )! . Here (cid:0) n k (cid:1) is the multinomial coefficient defined by (cid:18) n k (cid:19) = n ! k ! k ! . . . k j ! , where | k | := k + · · · + k j = n ,and N ( k , i ) is the number of times the integer i appears in the j -tuple k ( i ∈ N ∗ and k ∈ ( N ∗ ) j ).For example, the coefficient C , , of the term g (4) ( g ′ ) when looking at the 6-th order derivativeof f ◦ g is C , , = 12! 6!4! · ·
1! = 15and the coefficient C , , , , , of the term g (4) g ′′ ( g ′ ) in the 10-th derivative is C , , , , , = 14! 10!4! · · · · ·
1! = 3150 . The difference between our formula and the Fa`a di Buno formula is that we use a simplersummation order and do not consider exponents of the form b j = 0. In particular, we do not needsummation over those ( b , . . . , b n ) satisfying (the difficult to grasp) condition P ni =1 ib i = n . Thatthese two formulas are equivalent though, immediately follows from a direct comparison of thecoefficients. Indeed, for fixed j and k = ( k , . . . , k j ) ∈ M j , | k | = n , we have: C n k = (cid:18) n ! k ! · · · k j ! (cid:19) N ( k , · · · N ( k , n )! = (cid:18) n ! 1 i ! · · · i ! | {z } b i times · · · i ℓ ! · · · i ℓ ! | {z } b iℓ times (cid:19) b i ! · · · b i ℓ != n ! b ! · · · b n ! n Y i =1 (cid:18) i ! (cid:19) b i where the b i m are those exponents that are different from zero and where k has been representedin the canonical form k = ( i , . . . , i | {z } b i times , · · · , i ℓ , . . . , i ℓ | {z } b iℓ times ) in decreasing order. Note that n X i =1 ib i = ℓ X s =1 i s b i s = j X p =1 k p = n and that n X i =1 b i = b i + · · · + b i ℓ = j. Next I would like to present the (intuitive) steps that led me to the discovery of the formula (1.1)above, at pre-PC times; the first (non-programmable) slide rule calculator SR50 had just appeared.1) I calculated explicitely the derivatives ( f ◦ g ) ( n ) up to the order 10 and wrote them down ina careful chosen order (see figure 1);2) An immediate guess is that ERIVATIVES 3 ( f ◦ g ) ( n ) ( x ) = n X j =1 f ( j ) ( g ( x )) (cid:18) X k ∈ M j | k | = n c n k g ( k ) ( x ) (cid:19) , for some coefficients c n k to be determined.3) Next I gave an inductive proof that this representation is correct; that needs the main stepof the construction: where does the factor g ( k ) · · · g ( k j ) with k + · · · + k j = n + 1 comes from? Solet us look at the ordered j -tuple ( k , . . . , k j ). This tuple is generated, through anti-differentiation,by the j j -tuples( k − , k , . . . , k j ) , ( k , k − , k , . . . , k j ) , · · · · · · , ( k , . . . , k j − (cid:16) g ( k − g ( k ) · · · g ( k j ) (cid:17) ′ = g ( k ) g ( k ) · · · g ( k j ) + · · · , (cid:16) g ( k ) g ( k − · · · g ( k j ) (cid:17) ′ = g ( k ) g ( k ) · · · g ( k j ) + · · · , etc.The main difficulty being that the components k j are not pairwise distinct. So their ”multiplic-ities” had to be taken into account. This lead to the guess that one may have c = 1 c n +1 k = j X i =1 N ( k − e ji , k i − N ( k , k i ) c n k − e ji where for i = 1 , . . . , j , e ji = (0 , . . . , , |{z} i − th , , . . . , e ji ∈ N j , k ∈ M j , | k | = n + 1 and 1 ≤ j ≤ n + 1.Note that the j -tuple k − e ji is not necessarily represented in the canonical form with decreasingcoordinates. Also, if the i -th coordinate of k is one, then the i -th coordinate of k − e ji is 0 and weidentify k − e ji with the associated ( j − k = (3 , , ,
1) we have20 = C , , , = 11 C , , , + 13 C , , , + 13 C , , , + 13 C , , , , where (3 , , , , (3 , , ,
1) and (3 , , ,
0) are identified with (3 , , c n k , | k | = n . Now there are (cid:0) n k (cid:1) = n ! k ! ...k j ! ways tochoose k objects out of n , then k objects of the remaining ones, and so on. Due to the multiplicty,one has again to divide by N ( k , i )!.This gives the guess that c n k = (cid:18) n k (cid:19)Y i N ( k , i )! .
5) These coefficients c n k actually satisfy the recursion relation above. Since c = 1, and the factthat the recursion relation determines uniquely the next coefficients, we are done: C n k = c n k . RAYMOND MORTINI f ′ g ′ f ′′ g ′ f ′ g ′′ f (3) g ′ f ′′ g ′′ g ′ f ′ g (3) f (4) g ′ f (3) g ′′ g ′ f ′′ g (3) g ′ f ′′ g ′′ f ′ g (4) f (5) g ′ f (4) g ′′ g ′ f (3) g (3) g ′ f (3) g ′′ (2) g ′ f ′′ g (4) g ′ f ′′ g (3) g ′′ f ′ g (5) f (6) g ′ f (5) g ′′ g ′ f (4) g (3) g ′ f (4) g ′′ (2) g ′ f (3) g (4) g ′ f (3) g (3) g ′′ g ′ f (3) g ′′ f ′′ g (5) g ′ · · · f (7) g ′ f (6) g ′′ g ′ f (5) g (3) g ′ f (5) g ′′ (2) g ′ f (4) g (4) g ′ f (4) g (3) g ′′ g ′ f (4) g ′′ g ′ f (3) g (5) g ′ · · · f (8) g ′ f (7) g ′′ g ′ f (6) g (3) g ′ f (6) g ′′ (2) g ′ f (5) g (4) g ′ f (5) g (3) g ′′ g ′ f (5) g ′′ g ′ f (4) g (5) g ′ · · · f (9) g ′ f (8) g ′′ g ′ f (7) g (3) g ′ f (7) g ′′ (2) g ′ f (6) g (4) g ′ f (6) g (3) g ′′ g ′ f (6) g ′′ g ′ f (5) g (5) g ′ · · · f (10) g ′ f (9) g ′′ g ′ f (8) g (3) g ′ f (8) g ′′ (2) g ′ f (7) g (4) g ′ f (7) g (3) g ′′ g ′ f (7) g ′′ g ′ f (6) g (5) g ′ · · · Further formulas and questions
Applying formula (1.1) for the function f ( x ) = log x, x > g ( x ) = e x gives n X j =1 ( − j − ( j − X k ∈ M j | k | = n C n k = 0 , n ≥ f ( x ) = x n and g ( x ) = e x one obtains n X j =1 (cid:18) nj (cid:19) j ! X k ∈ M j | k | = n C n k = n n . In particular, L := X k : | k | = n C n k ≤ n n . Is there an explicit expression for L ? If one uses f ( x ) = g ( x ) = e x , then (cid:0) e e x (cid:1) ( n ) | x =0 = eL. One may also ask the following questions:(1) What is X k ∈ M j | k | = n C n k (1 ≤ j ≤ n )?(2) What is max { C n k : | k | = n } ?(3) Is there a formula for the number of partitions of n with fixed length j ?In our scheme (figure 1), one can give easy formulas for the coefficients in each column. In fact,each element in a fixed column is a multiple of the first coefficient. More precisely, if k ≥ k ≥ k j >
1, then we have: C n + ℓ ( k ,...,k j , , . . . , | {z } ℓ − times ) = C n ( k ,...,k j ) (cid:18) n + ℓℓ (cid:19) , ℓ ∈ N . We observe that several columns coincide; for example C , , = C , and so the elements ofthe associated columns are the same.There are actually infinitely many pairs of colums that coincide (just use that C i (2 , , ,i ) = C i (4 , ,i ) for every i ≥ ERIVATIVES 5
Are there triples (or higher number) of columns that coincide?
Acknowledgements
I thank J´erˆome No¨el for his help in finding the exact TEX-commands forcreating the tabular form above. I also thank Claude Jung for valuable discussions around 1980.
References [1] Craik, Alex. Prehistory of Fa`a di Bruno’s formula, Amer. Math. Monthly 112 (2005), 119-130.[2] Johnson, Warren. The curious history of Fa`a di Bruno’s formula, Amer. Math. Monthly 109 (2002), 217-234.
Universit´e de Lorraine, D´epartement de Math´ematiques et Institut ´Elie Cartan de Lor-raine, UMR 7502, Ile du Saulcy, F-57045 Metz, France
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