GENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT

Transcription

1 Inerna J Mah & Mah Sci Vol 4, No 7 000) S Hindawi Publishing Corp GENERALIZATION OF THE FORMULA OF FAA DI BRUNO FOR A COMPOSITE FUNCTION WITH A VECTOR ARGUMENT RUMEN L MISHKOV Received January 999) Absrac The paper presens a new eplici formula for he nh oal derivaive of a composie funcion wih a vecor argumen The well-known formula of Faa di Bruno gives an epression for he nh derivaive of a composie funcion wih a scalar argumen The formula proposed represens a sraighforward generalizaion of Faa di Bruno s formula and gives an eplici epression for he nh oal derivaive of a composie funcion when he argumen is a vecor wih an arbirary number of componens In his sense, he formula of Faa di Bruno is is special case The mahemaical ools used include differenial operaors, polynomials, and Diophanine equaions An eample is shown for illusraion Keywords and phrases Differeniaion heory, compuaional mehods, classical combinaorial problems 000 Mahemaics Subjec Classificaion Primary 58C0; Secondary 7-08, 05A Inroducion Someimes, when analyzing nh order nonlinear sysems or developing mahemaical processes such as presening he soluion of a problem in he general nh order case, i is necessary o have an eplici formula for he nh derivaive of a composie funcion wih a vecor argumen whose componens are arbirary in number A powerful ool for managing nh order asks is he formula of Faa di Bruno giving an eplici epression for he nh oal derivaive of he scalar composie funcion f))wih a scalar argumen )Iff,, and are scalars and f)) is a composie funcion for which all he necessary derivaives are defined, hen he nh derivaive of he funcion f wih respec o, in accordance wih he formula of Faa di Bruno, is f n) = n!f k) ) ) k ) ) k n)) k n, ) k!k! k n!!) k!) k n!) k n where k = k k k n and he sum is over all nonnegaive ineger soluions of he Diophanine equaion k k nk n = n The following noaion is used in ): f n) -nh derivaive of he funcion f wih respec o, f k) -kh derivaive of he funcion f wih respec o, i) -ih derivaive of he funcion wih respec o The righ-hand sides of ) for n =,,3, are known as Bell polynomials Bell []) The formula ) and he Bell polynomials show ha any oal derivaive f n) is a linear funcion of he parial derivaives f k), k=,,,n, )

2 48 RUMEN L MISHKOV which are muliplied by he srucural coefficiens n!, 3) k!k! k n!!) k!) k n!) k n and he nonlinear funcions of he derivaives i),i=,,,n, ) ) k ) ) k n)) k n 4) There are various proofs of formula ) based on differen approaches Taylor series represenaion Jordan [4]), Bell polynomials Come [3]), umbral calculus Riordan [5] and Roman [6]) The applicaion of formula ) is in asks where an eplici formula for he nh derivaive of he composie funcion wih a scalar argumen f)) has o be used, so i is resriced o he cases when he argumen ) is a scalar The fac ha he formula of Faa di Bruno considers only one scalar argumen ) in he epression for he nh derivaive of he composie funcion f)) provokes he necessiy of a new formula which considers he case when ) is a vecor The objecive of his paper is o derive, prove, and presen such a new formula for he nh oal derivaive of he composie funcion f)) when is argumen ) = [ ), ),, r )] T is a vecor wih an arbirary number of componens Generalized formula for he nh derivaive of a composie funcion wih a vecor argumen This secion inroduces he main resul in he form of a heorem and develops he generalized formula on he basis of Faa di Bruno s formula wih he help of a specific approach Theorem If f and are scalars, ) = [ ), ),, r )] T is an r -vecor and f)) is a composie funcion, for which all he necessary derivaives are defined, hen D n f ) ) = n! n n r 0 n i!) k i q ij! n i= i) i= ) qi i) i= j= ) qi i) r ) qir, k f p p pr r ) where he respecive sums are over all nonnegaive ineger soluions of he Diophanine equaions, as follows k k nk n = n, 0 q q q r = k, q q q r = k, ) n q n q n q nr = k n,

3 GENERALIZATION OF THE FORMULA OF FAA DI BRUNO 483 and he differenial operaor D = d/d, p j he order of he parial derivaive wih respec o j, and k he order of he parial derivaive are p j = q j q j q nj, j =,,,r, k = p p p r = k k k n 3) Proof If he noaion D i = i) is inroduced wih = / and ) i = i / i hen, since always k = k k k n he operaor for parial differeniaion can be represened as k / k = k / k k k n and herefore i can be insered in he parenheses, so formula ) can be wrien as follows: or f n) = n! )) k )) k n)) kn f 4) k!k! k n!!) k!) k n!) k n = n!d k Dk Dkn 5) k!k! k n!!) k!) k n!) k n f n) The difference beween his record and ) is ha he parial derivaives ) and he nonlinear funcions 4) remain hidden in he operaors D k i i and are no shown eplicily In accordance wih formula ), he sequenial derivaives f i),i=,,3, are funcions wih an increasing number of argumens n f f ) = f ),ẋ), f ) = f ),ẋ,ẍ), f 3) = f 3),ẋ,ẍ, ), f n) = f n),ẋ,ẍ,, n) ) 6) The sequenial derivaives f i),i=,,3, can be obained by considering 6) and applying he rule of he firs oal derivaive of a composie funcion wih an arbirary number of argumens: f ) = ẋ ) f = D f, f ) = ẋ ẋẍ ) f ) = [ ẋ) ẍ ] f = D D ) f, f 3) f 4) = ẋ ẋẍ ẍ ) f ) = [ ẋ) 3 3 ẋ ẍ ] f = D 3 3D D D 3 )f, = ẋ ẋẍ ẍ 4)) f 3) = [ ẋ ) 4 6 ẋ) ẍ 4 ẋ 3 ẍ ) 4)] f = D 4 6D D 4D D 3 3D D 4) f, 7)

4 484 RUMEN L MISHKOV If he hird derivaive f 3) is developed in deail, hen he epressions ẋẍ ẍ ) f, ẍ ẋ)f 8) will be obained They give null resuls because he funcion f))does no depend on ẋ and ẍ respecively Such cases of dropping ou of some erms from he final resul are observed in he fourh and he ne derivaives As menioned in he inroducion, i is proved in a number of ways, ha he general descripion of hese derivaives is given by he formula of Faa di Bruno ) Now, le be he r -vecor ) = [ ), ),, r ) ] T, 9) respecively, = [ ] T =,,, 0) r be a vecor differenial operaor and f)) a composie funcion for which all he necessary derivaives are defined The derivaives f i),i=,,3, can be obained via a sequenial applicaion of he rule of he firs oal derivaive of a composie funcion wih an arbirary number of argumens, subsiuion of he previous derivaive and considering he number of is argumens: f ) = T ẋ) f = D f, f ) = T ẋ Ṫ ẍ) f ) = [ T ẋ) T ẍ ] f = D D ) f, f 3) f 4) = T ẋ Ṫ ẍ T ẍ ) f ) = [ T ẋ)3 3 T ẋ T ẍ T ] f = D 3 3D ) D D 3 f, = T ẋ Ṫ ẍ T ẍ Ṭ 4)) f 3) = [ T ẋ) 4 6 T ẋ ) T ẍ4 T ẋ T 3 T ẍ) T 4)] f = D 4 6D D 4D D 3 3D D 4) f, ) The process of deriving he derivaives ) is consubsanial wih ha of he derivaives 7), and hey have he same operaor form, neverheless in one case he operaors are scalar and in he oher vecor Tha is why heir general form can be wrien ou in accordance wih he formula of Faa di Bruno as f n) = n! T )) k T )) k T n)) kn f k!k! k n!! ) k ) k! n! ) k n ) This sep is valid because The derivaives ) are obained via he sequenial applicaion of he rule of he firs oal derivaive of a composie funcion wih an arbirary number of argumens The process of developing he derivaives ) is consubsanial wih he process of developing he derivaives 7) The formula ) is a vecor form of he nh derivaive because he operaors for

5 GENERALIZATION OF THE FORMULA OF FAA DI BRUNO 485 parial differeniaion are in a vecor form Hence, he parial derivaives and he nonlinear funcions of he derivaives i),i=,,,nare no shown eplicily To obain he eplici componen form of he formula for he nh derivaive of f)) i is necessary o raise he polynomials T i) ) k i,i=,,,n o he corresponding powers and hen o muliply hem For his purpose, i is suiable o number he polynomials in accordance wih he derivaive which hey conain The polynomial conaining ) is he firs, he one including ) is he second and so on Then for he ih polynomial i can be wrien ou i) i) r r i) ) ki = k i! q i!q i! q ir! i) ) qi i) ) qi r i) r ) qir 3) The sum is over all nonnegaive ineger soluions of he Diophanine equaion q i q i q ir = k i Here, he powers q ij are numbered wih inde i-number of he polynomial and j-number of he componen from he vecor If he n polynomials in he form 3) are muliplied, hen he resul is f n) = 0 n k f p p pr r n!k!k! k n! k!k! k n!!) k!) k n!) k n q!q! q r!q!q! q r! q n!q n! q nr! ) ) q )) q n) ) qn )) q )) q n) ) qn r ) ) qr r ) ) qr r n) ) qnr 4) or in concise record f n) = 0 n! n n r n i!) k i q ij! n i= i) i= ) qi i) i= j= ) qi i) r ) qir, k f p p pr r 5) whereas he sums are over all nonnegaive ineger soluions of he Diophanine equaions k k nk n = n, 0 q q q r = k, q q q r = k, 6) n q n q n q nr = k n,

6 486 RUMEN L MISHKOV wih p j being he order of he parial derivaive wih respec o j and k he order of he parial derivaive p j = q j q j q nj, j =,,,r, k = p p p r = k k k n 7) This resul coincides wih formula ), which complees he proof Consisency Formula ) is an eplici componen form of he nh derivaive of he composie funcion f)) I is a sraighforward generalizaion of he formula of Faa di Bruno when he argumen ) is a vecor In his sense, Faa di Bruno s formula is a special case of formula ) when r =, and herefore ) = ) Then, he Diophanine equaions ) and he variables 3) ake he form k k nk n = n, 0 q = k, q = k, n q n = k n, p = q q q n, k = p = k k k n 8) In his case, he epression ) reads D n f ) ) = 0 n! k f k!k! k n!!) k!) k n!) k n k ) ) k ) ) k n) ) kn, 9) which is acually he formula of Faa di Bruno when = The use of formula ) requires he soluion of he sysem of linear Diophanine equaions ) When he order of he derivaive considered is low and he number of argumens r is small, hen hese equaions can be easily solved manually I is well known ha hey do no have general analyic soluion So, when we deal wih higher order derivaives and a bigger number of argumens, an efficien ool for solving such equaions is needed Clausen and Forenbacher [] have proposed algorihms and an efficien PASCAL program for solving a wide class of linear Diophanine equaions, including equaions of he ype ) The soluion of he sysem of linear Diophanine equaions can be obained smoohly wih he help of his program 3 Eample The fourh derivaive of he funcion f ), ) ) = e ) ) 3)

7 GENERALIZATION OF THE FORMULA OF FAA DI BRUNO 487 will be considered as an illusraion eample for formula ) Firs, he fourh derivaive of f ), )) will be considered in order o show is general srucure, and hen i will be applied over he funcion e ) ) In his case, formula ) has he form f ), ) ) 4) = i!) k i q ij! i= i= j= k f p p 4 i= i) ) qi i) ) qi 3) and he equaions ), k k 3k 3 4k 4 = 4, 33) 0 q q = k, q q = k, q 3 q 3 = k 3, 3 q 4 q 4 = k ) Table 3 No k k k 3 k The soluions of equaion 33) are shown in Table 3, and hey form he mos eernal loop of he sums These five soluions define he respecive five ses of combinaions of soluions of he equaions 34) as shown in Table 3 The five soluions of he main Diophanine equaion 33), corresponding o 0, are lised in he firs column of his able Each of hose soluions defines he Diophanine equaions 34), corresponding o i, i =,,3,4, which are shown in he second column All possible combinaions of soluions of he laer equaions are shown in he remaining columns The sums p = q q q 3 q 4 and p = q q q 3 q 4 deermine he order of he parial derivaives wih respec o and So, he fourh oal derivaive of he funcion f ), )) can be wrien ou in general form on he basis of formula 3) and Table 3,

8 488 RUMEN L MISHKOV Table 3 Soluions of he sysem of Diophanine equaions and heir combinaions 0 k k 3k3 4k4 = 4 i, i=,,3,4 Combinaion Combinaion Combinaion 3 Combinaion 4 Combinaion 5 Combinaion 6 q q = 4 q = 4 q = 0 q = q = 3 q = q = q = 3 q = q = 0 q = 4 q q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 k =4,k =0,k3 =0,k4 =0 3 q3 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 k = k k k3 k4 = 4 4 q4 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 k k 3k3 4k4 = 3 k = p p = 4 p = 4 p = 0 p = p = 3 p = p = p = 3 p = p = 0 p = 4 q q = q = q = 0 q = q = 0 q = 0 q = q = 0 q = q = q = q = q = q q = q = q = 0 q = 0 q = q = q = 0 q = 0 q = q = q = 0 q = 0 q = k =,k =,k3 =0,k4 =0 3 q3 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 k = k k k3 k4 = 3 4 q4 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 k k 3k3 4k4 = k = p p = 3 p = 3 p = 0 p = p = p = p = p = 0 p = 3 p = p = p = p = q q = q = q = 0 q = q = 0 q = 0 q = q = 0 q = q q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 k =,k =0,k3 =,k4 =0 3 q3 q3 = q3 = q3 = 0 q3 = 0 q3 = q3 = q3 = 0 q3 = 0 q3 = k = k k k3 k4 = 4 q4 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 k k 3k3 4k4 = k = p p = p = p = 0 p = p = p = p = p = 0 p = q q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q = 0 q q = q = q = 0 q = 0 q = q = q = k =0,k =,k3 =0,k4 =0 3 q3 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 k = k k k3 k4 = 4 q4 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 q4 = 0 k k 3k3 4k4 = k = p p = p = p = 0 p = 0 p = p = p = q q = 0 q = 0 q = 0 q = 0 q = 0 q q = 0 q = 0 q = 0 q = 0 q = 0 k = 0,k = 0,k3 = 0,k4 = 3 q3 q3 = 0 q3 = 0 q3 = 0 q3 = 0 q3 = 0 k = k k k3 k4 = 4 q4 q4 = q4 = q4 = 0 q4 = 0 q4 = k = p p = p = p = 0 p = 0 p =

9 GENERALIZATION OF THE FORMULA OF FAA DI BRUNO 489 f ), ) ) 4) = 4 f!) 4!) 0 3!) 0 ) 0 0!0!0!0!0!0!0! 4 ẋ 4 4 f!) 4!) 0 3!) 0 ) 0!0!0!0!3!0!0!0! 3 ẋ ẋ 3 4 f!) 4!) 0 3!) 0 ) 0!0!0!0!!0!0!0! 4 f!) 4!) 0 3!) 0 ) 0 3!0!0!0!!0!0!0! 3 4 f!) 4!) 0 3!) 0 ) 0 0!0!0!0!0!0!0! 4 ẋ 4 3 f!)!) 3!) 0 ) 0!!0!0!0!0!0!0! 3 ẋ ẍ 3 f!)!) 3!) 0 ) 0!0!0!0!0!!0!0! ẋ ẋ ẋ 3 ẋ ẋ ẍ 3 f!)!) 3!) 0 ) 0 0!!0!0!!0!0!0! ẍ ẋ 3 f!)!) 3!) 0 ) 0 0!0!0!0!!!0!0! 3 ẋ ẍ 3 f!)!) 3!) 0 ) 0!!0!0!!0!0!0! ẋ ẍ ẋ 35) 3 f!)!) 3!) 0 ) 0!0!0!0!!!0!0! ẋ ẋ ẍ f!)!) 0 3!) ) 0!0!!0!0!0!0!0! ẋ f ẋ!)!) 0 3!) ) 0!0!0!0!0!0!!0! f!)!) 0 3!) ) 0 ẋ 0!0!!0!!0!0!0! f!)!) 0 3!) ) 0 0!0!0!0!!0!!0! f!) 0!) 3!) 0 ) 0 0!!0!0!0!0!0!0! ẋ ẍ f!) 0!) 3!) 0 ) 0 0!0!0!0!0!!0!0! ẍ f ẍ!) 0!) 3!) 0 ) 0 ẍ 0!!0!0!0!!0!0! f 4)!) 0!) 0 3!) 0 ) 0!0!0!!0!0!0!0! f 4)!) 0!) 0 3!) 0 ) 0!0!0!0!0!0!0!!

10 490 RUMEN L MISHKOV or finally f ), ) ) 4) = 4 f 4 4 f 4 f 4 f ẋ ẋ ẋ 3 6 ẋ ẋ f 4 ẋ f f 3 3 f 3 f ẋ 3 ẋ ẋ ẍ 6 ẋ ẍ 6 ẍ ẋ 3 f 3 f ẋ ẍ ẋ ẍ ẋ ẋ ẋ ẍ 4 f ẋ 4 f ẋ 4 f ẋ 4 f ẋ 3 f ẍ 3 f ẍ 6 f ẍ ẍ f 4) f 4) 36) Specifically, for he funcion e ) ) he resul is e ) 4) = e [ 4 ẋ )ẋ ẋ ) ẋ ẋ )ẋ3 ẋ 4 ẋ4 63 ẋ ẍ 6 )ẋ ẍ 6 )ẍ ẋ 63 ẋ ẍ )ẋ ẍ ẋ )ẋ ẋ ẍ 4 ẋ 4 ) ẋ 4 ) ẋ 4 ẋ 3 ẍ 3 ẍ 6 ) ] ẍ ẍ 4) 4) 37) A comparison has been made wih he resul obained from he REDUCE program for symbolic compuaions wih he purpose of checking i up I has given he same epression for he fourh oal derivaive of he funcion e ) ) 4 Conclusions The formula proposed in his paper is an eplici componen form of he nh oal derivaive of a composie funcion of a vecor argumen wih an arbirary number of componens I is a sraighforward generalizaion of he formula of Faa di Bruno, and i is consisen wih he laer, indeed In his sense, he formula of Faa di Bruno is a special case of formula ) and can be obained from i when r = and herefore ) = ) The number of nesed sums in he new formula is n agains in formula ) which corresponds o he number of Diophanine equaions The greaer compleiy is he cos of he abiliy o manage composie funcions of a mulivariable argumen References [] E T Bell, Pariion polynomials, Ann Mah 9 97), [] M Clausen and A Forenbacher, Efficien soluion of linear Diophanine equaions, J Symbolic Compu 8 989), no -, 0 6 MR 90i:5 Zbl [3] L Come, Advanced Combinaorics, D Reidel Publishing Co, Dordrech, 974 MR 57#4 Zbl [4] C Jordan, Calculus of Finie Differences, Chelsea Publishing Co, New York, 965 MR 3#463 Zbl

11 GENERALIZATION OF THE FORMULA OF FAA DI BRUNO 49 [5] J Riordan, An Inroducion o Combinaorial Analysis, John Wiley & Sons Inc, New York, 958 MR 0#3077 Zbl [6] S Roman, The formula of Faa di Bruno, Amer Mah Monhly ), no 0, MR 8d:6003 Zbl Rumen L Mishkov: Sr Asen Zlaarov 3A, Plovdiv 4000, Bulgaria address: roumen@nevisione

12 Advances in Operaions Research Volume 04 Advances in Decision Sciences Volume 04 Applied Mahemaics Algebra Volume 04 Probabiliy and Saisics Volume 04 The Scienific World Journal Volume 04 Inernaional Differenial Equaions Volume 04 Volume 04 Submi your manuscrips a Inernaional Advances in Combinaorics Mahemaical Physics Volume 04 Comple Analysis Volume 04 Inernaional Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Mahemaics Volume 04 Volume 04 Volume 04 Volume 04 Discree Mahemaics Volume 04 Discree Dynamics in Naure and Sociey Funcion Spaces Absrac and Applied Analysis Volume 04 Volume 04 Volume 04 Inernaional Sochasic Analysis Opimizaion Volume 04 Volume 04