A DISCRETE VIEW OF FAÀ DI BRUNO. 1. Introduction. How are the following two problems related?

Transcription

1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number, 06 A DISCRETE VIEW OF FAÀ DI BRUNO RAYMOND A BEAUREGARD AND VLADIMIR A DOBRUSHKIN ABSTRACT It is shown how solving a discrete convolution problem gives a unique insight into the famous Faà di Bruno formula for the nth derivative of a composite function Introduction How are the following two problems related? Problem The chain rule for differentiating a composite function fgx is familiar Finding the nth derivative of fgx is more problematic but can be achieved using the classic Faà di Bruno formula: d n dx n fgx n! a! a! a n! f k gx g a x g a x g n an x,!! n! where the sum is taken over all nonnegative integer solutions a, a,, a n of the equation a + a + + n a n n, and where k a + a + + a n Johnson [5] takes the reader through the historical journey as this formula found its way into real analysis, combinatorial analysis, matrix methods and various other applications Problem Given two sequences of numbers or polynomials k {k n } n 0 and r {r n } n 0, the expression n n u n k n j r j k j r n j j0 is called the convolution of these two sequences and is denoted by u k r An interesting task presents itself when one attempts to solve, for the sequence r, the convolution equation u r r, where u 00 AMS Mathematics subject classification Primary B37, B83, 39A Keywords and phrases Discrete convolution, multinomial coefficient Received by the editors on April 3, 04 DOI:06RMJ Copyright c 06 Rocky Mountain Mathematics Consortium 73 j0

2 74 RA BEAUREGARD AND VA DOBRUSHKIN is a given sequence That is, we want to find the r j in terms of the u i in the system of equations u n n r n j r j, n 0,,, j0 This convolution problem arises naturally in various settings [, 4, 7] The most immediate application is illustrated with the sequence of Legendre polynomials which, when convoluted with itself, results in the sequence of Chebyshev polynomials of the second kind Our goal is twofold: a to show how the discrete convolution solution provides a unique insight into the structure of the Faà formula, and b to see why Problem can be used to solve Problem The discrete convolution equation Consider two sequences {u n } n 0 and {r n } n 0 satisfying, where u 0 and r 0 We want to find the solution sequence {r n } n 0 when {u n } n 0 is given Equation can be solved recursively: u r 0 r + r r 0 r u, u r + r r r u r u 3 u, u 3 r 3 + r r r 3 u 3 u u + 4 u3, u 4 r 4 + r r 3 + r r 4 u 4 u u u u 3 u Similarly, we find 5 7 u4 r 5 u 5 u u 4 u u u u u u 5 5 u3 u u5, r 6 u 6 u u 5 u u 4 3 u 3

3 A DISCRETE VIEW OF FAÀ DI BRUNO 75 and so on u u u u u u3 5 5 u3 u u u u4 u 0 u6, These equations illustrate that each r n is expressed as a sum of products u a ua ua n n, where a i 0 and a + a + + n a n n, one summand for each partition of n Recall that a partition of a positive integer n is a representation of n as a sum of positive integers, where their order does not matter and the number of summands is only bounded by n However, it is common practice to write individual summands in nondecreasing order: 3 n }{{}}{{}}{{} n a a a n, where a k, k,,, n, is the number of times the integer k appears in the partition To illustrate, the summands for r 6 in equation reflect all partitions of 6, which we list respectively: For example, the partition +++3 of 6 corresponds to the summand 5 5 u 3 u 3, where a 3, a 3 and all other a s are zeros In order to describe our sum r n for general n, we follow Riordan [6] and adopt the following notation For each partition 3 of the integer n, we associate vector pn a, a,, a n For example, the partition of the number is associated with the -component vector 3,, 0, 0,, 0, 0, 0, 0, 0, 0, 0, which is shortened further to 3,, 0, 0, 3e +e +e 5, where e j is the unit coordinate vector whose components are all 0 except for the jth which is We express the intrinsic connection between two sequences {u n } and

4 76 RA BEAUREGARD AND VA DOBRUSHKIN {r n } that are related through the convolution formula as follows: 4 r n r n u, u,, u n B pn u pn, pn Pn where summation is taken over the set Pn of all partitions pn of the integer n, u pn u a ua ua n n, and B pn is its coefficient In our illustration of the partition of 6 above, u 3,0, u 3 u 3, and its coefficient in the sum representation of r 6 is B 3,0, B 3e +e To determine the coefficients B pn in 4, we rewrite the convolution equation as n u n r n + r k r n k, k which leads to the full-history recurrence: 5 r n n u n r k r n k, r 0, r u k This equation shows that r n depends on u n linearly, but the other components u,, u n appear in the formula for r n through summation of their products Since r n r n u, u,, u n depends only on the first n values of the sequence {u n } n, u 0, equation 5 has a unique solution which can be found recursively However, the amount of work and resources needed to do this grows as a rolling snowball We will eventually avoid the full-history recurrence and express the coefficients B pn in an explicit way Let us isolate u n in equation 4 and write 6 r n u n + pn B pn u pn, n,,, where u pn u a ua ua n n and summation is taken over all partitions pn of the number n except n itself We find from equations 5 and 6 that r n pn Pn B pn u pn n u n r k r n k k

5 A DISCRETE VIEW OF FAÀ DI BRUNO 77 u n n B pk u pk B pn k u pn k k pk pn k }{{}}{{} r k r n k Comparing like powers of u i, we obtain 7 B pn n B pk B pn k, k pk where the inner sum is taken over all partitions pk of the integer k 0 < k < n For example, we know that for r 6, B 3,0, B 3e+e This coefficient is evaluated as B 3,0, B,0, B + B,0, B + B 0,0, B 3 + B 3 B 0,0, because, from, + B B,0, + B B,0, 5 5 B,0, 3 4, B, B,0,, B 3, B 0,0,, B 3 4 Notice that vector entries in summands cannot exceed corresponding entries in the original vector, in this case, 3, 0, An interesting pattern of coefficients is hidden in the full-history recurrence 7, which is empirically obtained and readily verifiable Some particular coefficients can be determined immediately If n has a partition pn re j partitioned into r integers equal to j, then 8 B rej c 0 r, n jr, n,,, where the sequence {c 0 j } j is generated by the function + z c 0 j z j j z z 3 + z3 4 5 z z5 z6 33 z7 49 z

6 78 RA BEAUREGARD AND VA DOBRUSHKIN These binomial coefficients c 0 n n satisfy the full-history recurrence n 9 c 0 n+ c 0 i c 0 n i+, n, i which is a consequence of the Vandermonde convolution [] of the sequence {c 0 n } with itself, analogous to 7 Applied to the jth entry of the vector pn re j, this recurrence verifies 8 For example, the coefficient B 6 of u 6 is c 0 6 and is computed using either 7 or 9 as B 6 B B 5 + B B 4 + B 3 B 3 + B 4 B + B 5 B Actually, the coefficients B pn are products of c 0 j and various coefficients c k j, where {c k j } { k j } is the sequence of binomial coefficients generated by the function + z k, k,, We proceed to derive explicit expressions for coefficients B pn Generalizing the formula B rej c 0 r, the data that we generated suggest that for any distinct unit coordinate vectors e t and for any positive integers p, q, r, s, we have B qei+re j B pei +qe j +re k B sei +pe j +qe k +re l c r q c 0 r, c q+r p c r q c 0 r c p+q+r s c q+r p c r q c 0 r In general, the coefficients are given by the formula 0 B pn B n j a je ij c n j aj a n For example, if v e i + e j + e k + 3e t + e l, then B v c 6 c5 c4 c 3 c c a+a a 3 c a a c 0 a

7 A DISCRETE VIEW OF FAÀ DI BRUNO 79 This is a very revealing and appealing structural pattern of coefficients B pn Notice that the coefficient of each unit coordinate vector serves as a subscript of a term c, and the sum of a subscript and superscript in any term c is equal to the superscript in the preceding c Since the unit coordinate vectors in equation 0 can be permuted, we see that the symbols a i in the expression on the right-hand side of 0 can be permuted We give an independent proof of this Theorem The symbols a i in the formula c n j a j a n c a +a a 3 c a a c 0 a can be permuted Proof Any permutation of symbols a i in formula can be achieved by repeatedly transposing pairs of adjacent symbols It can be shown see proof below that c a i+b a j c b a i from which the theorem follows c a j+b a i c b a j, Proof of equation The equation c a+b d c b a rewritten using binomial coefficients: 3 a b b b d b d a a d We will show that each side of 3 is equal to a + d b a a + d Recall the notation a b aa a b + c a b+d c b d can be for bth falling factorial and the definition of the binomial coefficient a ab b b!

8 80 RA BEAUREGARD AND VA DOBRUSHKIN Looking at the left-hand side of 3, we compute d! a b d a b a b a b d +, a! b a b b a b + Therefore, a b b d a a! d! b b a b + a b a b a b d + a+d a + d! a + d a! d! a + d! b b a a + d The reasoning for the right-hand side of 3 is similar This completes the proof The identity we have established can be written as c a+b a + d d c b a c b a+d a We apply this identity successively to formula working from left to right The first two applications result in an + a n B pn c n i ai a a n +a n c n 3 i a ai n c a a c 0 a n an + a n an + a n + a n c n 4 i a i a a n a n + a n+a n +a n c a a c 0 a n an + a n + a n c n 4 i ai a a n, a n, a n +a n +a n c a a c 0 a n

9 A DISCRETE VIEW OF FAÀ DI BRUNO 8 Continuing this process, we obtain a product of n integer binomial coefficients, which simplifies to a multinomial expression, multiplied by the last factor c 0 n n We summarize our work as follows i ai i ai Theorem The coefficients 0 in the solution 4 of the convolution problem are given by n 4 B pn B n i a i i aiei n a, a,, a n i a, i where n i a ie i is associated with a partition of the integer n as described previously Proof Equation 8 gives the coefficient a j of u a i j when the partition of n ja j involves a j copies of a single integer j If the partition of n involves only two integers k and j, used a k and a j times respectively, then we can think of u a k k ua j j as a single entity containing a k +a j factors, which, by 8, has coefficient a k +a j, and there are a k +a j a k, a j such expressions This results in the total coefficient ak +a j a k, a j a k +a j for u a k k ua j j Using this reasoning, we arrive at 4 Although the partition of n includes all of the integers,, n, the frequency values a i are allowed to be zero Indeed, if a n or if a n, then a i 0 for all other i Example 3 Consider a coefficient B p corresponding to the vector p e i + 5e i + 4e i3 + e i4 Then B p c c 6 5 c 4 c , 5, 4, Example 4 Let us compute r 7 using the coefficient formula 4: r 7 u 7 + u u 6 + u u 5 + u 3 u 4, u u u 4 + u,, 3, 3 u 5 + u u 3 + u u 3

10 8 RA BEAUREGARD AND VA DOBRUSHKIN 4 +, 3 5 +, 3 6 +, u 3 u 4 + u u 3 5 u 3 u + u 5 u +, ,, 4 u 4 u 3 u 7 5 u u u 3 Each summand corresponds to exactly of the 5 partitions of n 7 For example, the summand 5,3 5 u 3 u corresponds to the partition ++++ Evaluating the coefficients, we obtain r 7 u 7 u u 6 + u u 5 + u 3 u u u u u u 5 + u u 3 + u u 3 5 u 3 5 u 4 + u u u u u u3 u u4 u u5 u + 33 u7 3 The nexus We reconcile the convolution solution 4, r n r n u, u,, u n B pn u pn, pn Pn with the Faà di Bruno formula The coefficient formula 4 in Theorem may be written as B pn B n k, i aiei a, a,, a n k where k n i a i The binomial factor k is the kth coefficient, equal to f k 0k!, in the Maclaurin expansion of fx + x Letting Uz n 0 u n z n, we see that u k U k 0k! for k Since u pn u a ua uan n, equation 4 becomes 3 r n k pn Pn f k 0 a,a,,a n k! U 0! a U 0! a U n an 0 n!,

11 A DISCRETE VIEW OF FAÀ DI BRUNO 83 where fuz Uz This matches the Faà di Bruno formula evaluated at x 0 and divided by n! Thus, we see how the sum in the Faà formula varies over all partitions of n when viewed through the discrete convolution solution r n in 3 See also the treatment by Flanders [3] The aforementioned function Uz n 0 u n z n is a generating function for the sequence u {u n } n 0 Letting Rz n 0 r n z n be the generating function for the sequence r {r n } n 0, we have Uz Rz, which corresponds to the convolution equation u r r Hence, r n is the coefficient of z n in the Maclaurin expansion of Uz For example, the generating function for the sequence {P n x} n 0 of Legendre polynomials is P x, z xz + z n 0 P n xz n, and P x, z generates the Chebyshev polynomials of the second kind [] REFERENCES CE Cook and M Bernfeld, Radar signals: An introduction to theory and application, Artech House, Norwood, MA, 993 VA Dobrushkin, Methods in algorithmic analysis, CRC Press, Boca Raton, 00 3 H Flanders, From Ford to Faà, Amer Math Month 08 00, J Goodman, Introduction to Fourier optics, Roberts and Company Publishers, Greenwood Village, CO, JP Johnson, The curious history of Faà di Bruno s formula, Amer Math Month 09 00, J Riordan, An introduction to combinatorial analysis, Wiley, New York, PK Suetin, Solution of discrete convolution equations in connection with some problems in radio engineering, Uspek Mat Nauk , 97 6 in Russian; Russian Math Surv , 9 43 in English University of Rhode Island, Kingston, RI 088 address: beau@mathuriedu University of Rhode Island, Kingston, RI 088 address: dobrush@mathuriedu