A generalization of the Leibniz rule for derivatives

Transcription

1 A geeralizatio of the Leibiz rule for erivatives R. DYBOWSKI School of Computig, Uiversity of East Loo, Docklas Campus, Loo E16 RD I will shamelessly tell you what my bottom lie is. It is placig balls ito boxes.... Gia-Carlo Rota (Iiscrete Thoughts) 1 Itrouctio It is commo kowlege that the first erivative of the prouct f(x)g(x) is give by f (x)g(x) + f(x)g (x), a that the seco erivative is f (x)g(x) + f (x)g (x) + f(x)g (x). We look at the more geeral case; amely, the -th erivative of a prouct of m fuctios f 1 (x) f m (x). Accorig to the Leibiz rule [e.g., 1, p. 534], the -th erivative of a prouct of two fuctios is give by f(x)g(x) = ( ) f (r) (x)g ( r) (x), (1) r r=0 where f () (x) eotes the -th erivative of fuctio f(x), with f (0) (x) = f(x), but what is the geeral form whe we have m fuctios: f 1(x) f m (x)? We will aswer this questio by usig the combiatorial tool of balls i boxes. Balls a boxes There are m ways of allocatig labele balls to m empty boxes. Each possibility will be referre to as a allocatio. The occupacy vector (α 1,..., α m ) eotes a allocatio havig α i balls (α i 0) i the i-th box. The umber of ways of allocatig α 1 labele balls i the 1st box, α labele balls i the box,..., α m labele balls i the m-th box is give by the multiomial coefficiet ( ) = α 1,..., α m! α 1! α m!, where α α m = ; thus, ( ) of the m possible allocatios have the occupacy vector (α 1,...,α m ). 1

2 Let b 1 b b m represet a allocatio of b 1 + b + + b m labele balls i m boxes, where b i is the set of labele balls i the i-th box. The occupatio vector correspoig to this allocatio is ( b 1, b,..., b m ). For example, {a, b} {c} is a allocatio base o three boxes (the seco box beig empty), a its correspoig occupacy vector is (, 0, 1). Let L 1 b 1 b m represet the set of allocatios resultig from the m possible ways of allocatig oe labele ball, say x, to the boxes of b 1 b m : For example, L 1 b 1 b m = b 1 {x} b m,..., b 1 b m {x}. () L 1 {a, b} {c} = { {a, b, x} {c}, {a, b} {x} {c}, {a, b} {c, x} }. We will exte the applicatio of L 1 to a set of γ allocatios {u 1,...,u γ }: For example, L 1 {u 1,..., v γ } = L 1 u 1 L 1 u γ. L 1{ {a}, {a} } = L 1 {a} L 1 {a} = { {a, b}, {a} {b} } { {b} {a}, {a, b} }. We ca use the L 1 operator to create the the set of all possible allocatios of labele balls i m boxes i a systematic, step-wise maer. Iitially, the m boxes are empty: m. The m possible ways of allocatig a labele ball to m is the set L 1 m. Aig a seco labele ball to the elemets of L 1 m i every possible way correspos to L 1 (L 1 m), but this is equal to the set of all possible ways of allocatig two labele balls to m (See Figure 1): Cotiuig i this maer, we obtai L m = L 1 (L 1 m). L m = L 1 (L 1 ( L 1 m )). (3) }{{} {a} {a} {a, b} {a} {b} {b} {a} {a, b} Figure 1: Formatio of possible elemets of L from allocatio via possible elemets of L 1.

3 .1 Multisets of occupacy vectors Let L 1 (α 1,..., α m ) eote the set (possibly multiset) of occupacy vectors resultig from firstly performig L 1 o a allocatio with occupacy vector (α 1,...,α m ) a the replacig each resultig allocatio with its correspoig occupacy vector. Put aother way, if a set of labele balls b i is such that b i = α i the L 1 (α 1,...,α m ) = L 1 ( b 1,..., b m ) For example from () a (4), we have = Γ(L 1 b 1 b m ). L 1 (0, 0, 0, 0) = {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}. Aalogous to the case with L 1, we will exte the applicatio of L 1 to a multiset of γ occupacy vectors {v 1,...,v γ }: This ca be rewritte as L 1 {v 1,...,v γ } = L 1 v 1 L 1 v γ. γ L 1 {v i } = (4) γ L 1 v i, (5) where eotes the aitive uio operator of multisets []. The operator L 1 ca be geeralize to L ; amely, L (α 1,..., α m ) is the multiset of occupacy vectors resultig from performig L o a allocatio with occupacy vector (α 1,..., α m ) a the replacig each resultig allocatio with its correspoig occupacy vector: Theorem 1 L (α 1,..., α m ) = L ( b 1,..., b m ) L 1 (α 1,..., α m ) = where δ ij is the Kroecker elta: δ ij = = Γ(L b 1 b m ). m (α 1 + δ 1j,...,α m + δ mj ), j=1 { 1 if i = j, 0 otherwise. Proof. Let b i be ay set of labele balls such that b i = α i, the L 1 (α 1,..., α m ) = Γ(L 1 b 1 b m ) from (4) = Γ{ b 1 {x} b m,..., b 1 b m {x} } from () = {Γ b 1 {x} b m,..., Γ b 1 b m {x} } from (5) = {( b 1 + 1,..., b m ),..., ( b 1,..., b m + 1)} = {(α 1 + 1,...,α m ),..., (α 1,...,α m + 1)} m = (α 1 + δ 1j,...,α m + δ mj ). j=1 (6) 3

4 A importat relatioship exists betwee L 1 a L 1, as show by the followig lemma. Lemma 1. If u is a allocatio the ΓL 1u = L 1 Γu. Proof. Let u = b 1 b m the L 1u = { b 1 {x} b m,..., b 1 b m {x} } ; therefore, ΓL 1 u = {( b 1 + 1,..., b m ),..., ( b 1,..., b m + 1)}. However, Γu = ( b 1,..., b m ); therefore, L 1 Γu = {( b 1 + 1,..., b m ),..., ( b 1,..., b m + 1)}. The ext lemma extes Lemma 1 so that sets of allocatios ca be iclue. Lemma. If S is a set of allocatios the ΓL 1 S = L 1ΓS. Proof. Let S = {u 1,..., u γ } the L 1 S = L 1 {u 1,...,u γ } = γ L 1 u i from (5); therefore, ΓL 1S = Γ γ L 1u i = γ ΓL 1u i. Now, ΓS = {Γu 1,..., Γu γ }; therefore, L 1 ΓS = L 1 {Γu 1,...,Γu γ } = γ L 1Γu i = γ ΓL 1 u i from Lemma 1. Lemma allows a versio of (3) for L to be establishe. Theorem L (0,..., 0) m = L 1 L 1 L }{{ 1 (0,..., 0) } m. Proof. L (0,...,0) m = ΓL m from (4) = ΓL 1 L 1 m from (3) = L 1 ΓL 1 L 1 m from Lemma = = L 1 L 1 Γ m from Lemma = L 1 L 1 (0,...,0) m From Theorem 1, Theorem a (5), we ow have the followig system (System 1) that geerates the elemets of the multiset L (0,...,0) m : L (0,...,0) m = L 1 L 1 L 1 (0,...,0) }{{} m System 1 where L 1 (α 1,...,α m ) = m j=1 (α 1 + δ 1j,...,α m + δ mj ) a L γ 1 {v i} = γ L 1v i, v i eotig a occupacy vector. This geeratio of elemets is illustrate i Figure. 4

5 (0, 0) (1, 0) (0, 1) (, 0) (1, 1) (1, 1) (0, ) Figure : Formatio of the elemets of multiset L (0, 0) from occupacy vector (0, 0) via the elemets of L 1 (0, 0). 3 Beyo the Leibiz rule I orer to see more clearly the lik betwee the -th erivative of f 1 (x) f m (x) a L (0,...,0) m, we will use a special otatio. The prouct f (α1) 1 (x) f m (αm) (x) will be writte as the erivative-orer tuple α 1,...,α m ; for example, the erivatio f(a) 1 (x)f (b) (x) = f(a+1) ca be writte more succictly as 1 (x)f (b) (x) + f(a) 1 (x)f (b+1) (x) a, b = a + 1, b + a, b + 1. Furthermore, usig this otatio, the -th erivative of f 1 (x) f m (x) ca be reefie as 0,..., 0 m = 0,..., 0 m. (7) }{{ } The sum rule of ifferetial calculus ca be writte as γ w i = where w j is a erivative-orer tuple. Lemma 3. α 1,...,α m = where δ ij is the Kroecker elta. γ w i, (8) m α 1 + δ 1j,..., α m + δ mj, j=1 Proof. α 1,...,α m = α 1 + 1, α,...,α m + α 1, α + 1,...,α m + + α 1, α,..., α m

6 Gatherig together (7), (8) a Lemma 3, we obtai the followig system (System ) that geerates the terms of 0,..., 0 m (See Figure 3): 0,...,0 m = 0,...,0 m }{{ } System where α 1,..., α m = m j=1 α 1 + δ 1j,...,α m + δ mj, a γ w i = γ w i, w i eotig a erivative-orer tuple. Lemma 4. There are ( ) allocatio vectors i L (0,...,0) m equal to (α 1,...,α m ). Proof. L (0,..., 0) m = ΓL m, a, as previously state early i Sectio, ( ) of the m possible allocatios i L m have occupacy vector (α 1,..., α m ). We ow have i place the material require to prove the mai goal of this paper; amely, the -th erivative of f 1 (x) f m (x). Theorem 3 0,..., 0 m = α 1+ +α m= α i 0 ( ) α α 1,..., α 1,...,α m m Proof. By ispectio, it is clear that System 1 a System are isomorphous, with L a ; therefore, sice ( ) of the elemets i multiset ( ) L (0,...,0) m are equal to (α 1,..., α m ) (Lemma 4), it follows that of the terms i series 0,..., 0 m are equal to α 1,..., α m. Theorem 3 ca be rewritte as f 1(x) f m (x) = α 1+ +α m= α i 0 ( 0, 0 ) f α1 α 1,..., α 1 (x) fαm m (x). m 1, 0 0, 1, 0 1, 1 1, 1 0, Figure 3: Formatio of the terms of 0, 0 from erivative-orer tuple 0, 0 via the terms of 0, 0. Compare with Figure. 6

7 Refereces [1] W. Kapla. Avace Calculus. Aiso-Wesley, Reaig, MA, 4th eitio, [] W.D. Blizar. Multiset theory. Notre Dame Joural of Formal Logic, 30(1):36 66,