A Pair of Operator Summation Formulas and Their Applications

Transcription

1 A Pair of Operator Suatio Forulas ad Their Applicatios Tia-Xiao He 1, Leetsch C. Hsu, ad Dogsheg Yi 3 1 Departet of Matheatics ad Coputer Sciece Illiois Wesleya Uiversity Blooigto, IL , USA Departet of Matheatics Dalia Uiversity of Techology Dalia 11604, P. R. Chia 3 College of Applied Scieces Beiig Uiversity of Techology Beiig 1000, P. R. Chia July 8, 009 Abstract Two types of sybolic suatio forulas are reforulated usig a extesio of Mulli-Rota s substitutio rule i [1], ad several applicatios ivolvig various special forulas ad idetities are preseted as illustrative exaples. Key words Delta operator, Beroulli uber, Catala uber, geeralized haroic uber, Stirlig ubers Matheatics subect Classificatio (000 65B10, 05A10, 05A15, 05A19, 65B99 1 Itroductio The recet paper [] by He, Hsu, ad Shiue has show that, as a applicatio of the substitutio rule based o Mulli-Rota theory of bioial eueratio (cf. [1], the sybolizatio of geeratig fuctios ay yield ore tha a doze sybolic suatio forulas ivolvig delta operator ad D. Here let us recall that (differece operator ad D (differetiatio operator together with E (shift operator are usually defied for all f(t C (the class of ifiitely differetiable real fuctios i R = (, via the relatios 1

2 f(t = f(t + 1 f(t, Df(t = d dt f(t = f (t, Ef(t = f(t + 1. Cosequetly they satisfy soe siple sybolic relatios such as E = 1 +, E = e D, = e D 1, D = log E = log(1 +, where the uity 1 serves as a idetity operator such that 1f(t = f(t. Also, for ay real or coplex uber α, we ay defie E α f(t = f(t + α with E 0 = D 0 = 0 = 1. I additio, a operator T is called a shift-ivariat operator (see, for exaple, [1] if it coutes with the shift operator E, i.e., T E α = E α T, where E α f(t = f(t + α ad E 1 E. Clearly, the differetiatio operator D ad the differece operator are shift-ivariat operators. A operator Q is called a delta operator if it is shift-ivariat ad Qt is a o-zero costat. Obviously, both D ad are delta operators. What we wish to show is that the two types of sybolic suatio forulas expaded i [] ay be reforulated usig a extesio of Mulli-Rota s substitutio rule so that they could apply to ore cases tha those give previously. Accordigly we will cosider soe ew applicatios, ad preset several exaples ad idetities ivolvig soe special uber sequeces such as Beroulli, Catala, Stirlig, haroic ubers ad the geeralized haroic ubers. I additio, we shall show that the foral power series ca be recovered fro the correspodig sybolic suatio forulas by substitutig a certai chose fuctio. Two basic theores Let Q be a delta operator, ad let F be the rig of foral power series i the variable t, over the sae field, the [1] proved that there exists a isoorphis fro F oto the rig of shift-ivariat operators, which carries g(x = 0 a! x ito g(q G(x, Q := a! Q. 0 The above rule is called Mulli-Rota s substitutio rule. Deote by G(x, y, z a ratioal fuctio i three variables x, y, ad z. I particular, G(x, y, 1 ad G(x, 1, 1 deote ratioal fuctios i two variables ad oe variable, respectively. I what follows we always assue that F (x =0 f x is a foral power series. The we shall use Mulli-Rota s substitutio rule to establish the followig results.

3 Theore.1 Suppose that for give power series F (x there is a expressio or a su forula of the for f x = G(x, e x, e αx, (.1 0 where the paraeter α 0 is a real or coplex uber. The the substitutio x D yields a sybolic suatio forula for every f C evaluated at t = 0, aely f D f(0 = G(D, E, E α f(0. (. 0 Moreover, (. iplies (.1 as a particular case with f(t = e xt. Theore. Suppose that for give power series F (x there is a expressio or a su forula of the for f x = G(x, log(1 + αx, (1 + αx β, (.3 0 where α ad β are real paraeters with αβ 0. The the substitutio x 1 α yields a sybolic suatio forula of the for ( 1 f f(0 = G( α α, D, Eβ f(0. (.4 0 Moreover, (.4 iplies (.3 as a particular case with f(t = (1 + αx t. Proof: Theores.1 ad. ca be proved siilarly. Sice both D ad are delta operators, so that (. ad (.4 as sybolizatios of (.1 ad (.3, respectively, ca be ustified by a siilar arguet of Mulli-Rota s substitutio rule (see [1] or []. More precisely, both (.1 ad (.3 are idetities i the variable x, ad that there is a isoorphis betwee the rig of shift-ivariat operators ad the rig of foral power series i x. Hece, (. ad (.4 are obtaied accordigly. It reais to show that the choices f(t f(t; x = e xt ad f(t f(t; x = (1 + αx t will respectively lead (. ad (.4 to recover (.1 ad (.3. For the particular choice f(t = e xt we see that the right-had side (RHS of (. ca be writte as follows RHS of (. = G(D, e D, e αd f(0 = 0 f D f(0 = 0 f D e xt t=0 = 0 f x = G(x, e x, e αx. Also, the left-had side (LHS of (. with f(t = e xt gives 0 f x. Hece, (.1 is iplied by (.. 3

4 The iplicatio (.4 (.3 with f(t = (1 + αx t ca be verified i a siilar aer, i which it suffices to observe that the LHS of (.4 with f(t = (1 + αx t gives 0 f x, ad that the RHS of (.4 gives ( G, log(1 +, (1 + β f(0 α f(0 = ] (1 + αx t = ( f α 0 0 f [ ( α = 0 f x = G ( x, log(1 + αx, (1 + αx β, t=0 which copletes the proof. The followig two exaples ay further illustrate the secod halves of the theores. First, usig (. with F (x = e ax = 0 (ax /! with x D yields the suatio forula 0 a D f(0/! = f(a, which iplies e ax = 0 (ax /! as a special case with f(t = e xt. Siilarly, if (.3 is give with F (x = log(1 x = 1 x /!, the the correspodig suatio forula (.4 with the appig x ( is 1 ( 1+1 f(0/ = f (0, which iplies 1 x /! = log(1 x as a special case with f(t = (1 x t. The techique preseted i the above theores ca be cosidered as extesios of (Mulli-Rota s substitutio rule. For brevity, forulas (. ad (.4 ay be siply called D-type forula ad -type forula respectively. These forulas obviously provide geeralizatios of the su forulas for sigle power series. As ay be observed, substatially all the operatioal forulas (O (O 1, as displayed i [], together with the sybolic forulas expressig D (or i ters of s (or D s are particular cosequeces of (. ad (.4, respectively. It ay be oted that the operatioal forula give i Exaple 5.14 of [] of the for (O 13 : +1 f( = ( 1 +1 A (Ef(0 =0 is icorrect, where A (x deotes the th degree Euleria polyoial give by the expressio with A(, 0 = 0 ad A 0 (x = 1 ad A (x = A(, = A(, x ( 1, =1 ( + 1 ( 1 ( (1. =0 4

5 A(, are ow to be the Euleria ubers (cf. Cotet [3, p. 43-5]. I fact, taig f(t to be a polyoial of degree with 1, we see that the LHS of (O 13 gives zero, while the RHS differs fro zero. Actually (O 13 is obtaied fro the sybolizatio of Euler s forula x = α (x = A (x ( x < 1, (1 x +1 =0 by the substitutig x E, where E = 1 + is ot a delta operator iasuch as Et = t + 1 is ot a o-zero costat. A valid sybolizatio should be ade by the substitutio x (, so that Euler s forula yields a special - type forula of the for (O 14 : ( 1 f(+1 = A ( f(0 = A(, ( 1 f(0. 0 =1 Taig f(t = 1/(1 + t ito (O 14, we fid (cf. (5.17 of [] 1 + =0 ( = + A(, /( + 1 ( 1. Curiously eough, this correct suatio is also obtaiable fro the icorrect forula (O 13. This ight suggest that (O 13 could still be valid uder certai restrictive coditios. Oe ay recover Euler s forula fro (O 14 by substitutig f(t = (1 x t. Ideed, for the fuctio f(t, we have f( + 1 = (1 x +1 ( x ad f(0 = ( x. Thus [A ( (1 x t ] t=0 = =1 A(, ( 1 f(0 = =1 A(, x = A (x, ad (O 14 becoes 0 (1 x +1 x = A (x, which is the Euler s forula 0 x = A (x/(1 x +1 for x 1. =1 3 Applicatio of (. ad (.4 I additio to those geeratig fuctios already ivestigated i [], let us ow cosider soe other geeratig fuctios or power series expasios with closed sus as follows (cf. Wilf [4]. (i 0 4 B (! x = x coth x, where B are Beroulli ubers. φ r(x φ r(0! x, where φ r (x is a rth degree polyo- (ii 0! = e x r =0 ial (cf. Jolley [5, p. 18]. ( 1 1 4x, where C = 1 +1( are Catala u- (iii 0 C x = 1 bers. x (iv 1 H x = 1 1 x log 1 1 x, where H are haroic ubers defied by H = =1 1/ for 1 with H 0 = 0. 5

6 (v 1 H 1x = 1 ( log 1 1 x (vi ( +r ( 0 x = 1 r 1 1 4x 1 4x x (r 0 (vii ( r+1 1 H(, rx = 1 1 x log 1 x 1, where H(, r are geeralized haroic ubers (cf. [6] defied by H(, r = 1 1/( r 0 1 r for 1 ad r 0 with H(0, r = 0. It is obvious that H(, 0 = H. (viii ( r(+r 1! 0!(+r! x r = 1 1 4x x, which icludes (iii as a special case whe r = 1. Evidetly, (i ad (ii are of the for (.1, ad (iii-(viii of the for (.3. Cosequetly (i ad (ii should lead to special D-type forulas, ad (iii-(viii to -type forulas. Ideed, aig use of (. we easily fid 4 B (! D f(0 = D E + E 1 f(0. E E 1 0 Notice that (E E 1 D f(0 = f ( (1 f ( ( 1. Thus we ca obtai a sybolic suatio forula of the for 0 4 B (! [f ( (1 f ( ( 1] = f (1 + f ( 1. (3.1 Siilarly, utilizig forulas (. ad (.4 oe ay fid that (ii-(viii yield 7 special sybolic suatio forulas as follows 0 0 φ r ( f ( (0 =! ( 1 4 r =0 C +1 f(0 = [ f φ r (0 f ( (1 (3.! ( ] 1 f(0 (3.3 ( 1 H f(0 = f ( 1 (3.4 1 ( 1 H 1 f(0 = 1 f (0 (3.5 ( 1 ( ( + r +r +r f(0 = (E 1/ 1 r f 1, (3.6 0 ( 1 H(, r f(0 = ( 1 r+1 f (r+1 ( 1, ( ( 1 r( + r 1! 4!( + r! +r f(0 = r ( r ( 1 r =0 f (, (3.8 6

7 where the RHS of (3.6 ay be writte i the explicit for (E 1/ 1 r f ( 1 = r ( r ( 1 r f =0 ( 1. (3.9 More precisely, (3.-(3.8 are obtaied fro (ii-(vi by the substitutios x D, x ( 1 4, x (, x ( 1 4, x (, ad x ( 1 4 respectively. 4 Soe covergece coditios Here we provide a list of coditios for the absolute covergece of the series expasios i (3.1-(3.8. forula covergece coditio (3.1 li f ( (±1 1/ < π (3. li f ( (0/! 1/ < 1 (3.3 f(0 = O ( 1 ɛ (ɛ > 0 (3.4 f(0 = O ( (1/ 1+ɛ (ɛ > 0 (3.5 f(0 = O ((1/ ɛ (ɛ > 0 (3.6 li f(0 1/ < 1 (3.7 f(0 = O ( (1/ 1+ɛ (ɛ > 0 (3.8 f(0 = O ( 1 ɛ (ɛ > 0 The covergece coditios show above ca be ustified by the aid of Cauchy s root test ad the copariso test. Notice that there is a estiate for Beroulli ubers, viz. (cf. Jorda [7, 8] B (! < 1 1(π ( 0. It follows that upper liit B li (! 7 1/ 1 4π

8 (Actually, Euler s faous forula for Beroulli ubers, ( 1 +1 B /(! = ζ(/(π iplies the liit of (B /(! 1/ equals to 1/(4π, so that the covergece coditio for (3.1 iplies that 4 B li (! f ( (±1 1/ < 1. Hece the absolute covergece of the series i (3.1 follows fro the root test. Moreover, otice that li φ r ( 1/ = 1 ad that li 1 +r ( + r 1/ = 1, where the liit follows fro a applicatio of Stirlig s asyptotic forula! (/e π as. Thus the covergece coditios for (3. ad (3.6 also follow fro the root test. Evidetly the covergece coditios for (3.3, (3.4, (3.5, (3.7, ad (3.8 are ustified by the followig asyptotic relatios, respectively C = 1 ( 4 /( π, + 1 H(, r log (r = 0, 1,..., r( + r 1! 4 3/,!( + r! as. Here, the secod estiatio for r 1 coes fro [6, (3.]. 5 Exaples- Various idetities ad series sus Certaily, each of the forulas (3.1-(3.8 ay be used to yield a variety of particular idetities or series sus via suitable choices of f(t. Here we will preset a uber of selective exaples to illustrate the applicatios of (3.1- (3.8. Exaple 1 Let be a odd positive iteger, ad tae f(t = t, ( 1. The we have f (1 = f ( 1 =, f ( (±1 = ±, where we use the followig fallig factorial otatio x r (soeties also deoted (x r, i.e., x r = x(x 1 r 1 (r 1 with x 0 = 1. Thus usig (3.1 we get [/] =0 ( 4 B =. (5.1 Exaple Let λ be a real uber with λ 0. The a uch ore geeral idetity of the for 8

9 ( ( ( λ + 1 λ + + λ B = λ + 1 =0 (5. ca be obtaied fro (3.1 by taig f(t = C λ (t with = +1, where C λ (t is the th degree Gegebauer polyoial give by the geeratig fuctio (1 tx + x λ = C λ (tx (λ 0. (5.3 =0 Ideed, a few siple properties of C λ (t ay be deduced fro (5.3, aely (cf. Magus-Oberhettiger-Soi [8, 5.3] C(1 λ (λ =, C λ! ( t = ( 1 C(t, λ ( d C λ dt (t = λ C (t, λ+ where we have used the raisig factorial otatio x r (soeties also deoted (x r or x r, i.e., x r = x(x + 1 r 1 (r 1 with x 0 = 1. Cosequetly, the fact that (3.1 iplies (5. is cofired by easy coputatios with the aid of the above etioed properties. For the particular choices λ = 1 ad λ = 1/, we see that (5. gives the followig idetities respectively ( ( B = ( ( ( B = 4 =0 =0, (5.4. (5.5 Exaple 3 Recall that Stirlig ubers of the first ad secod id ay be defied by the followig equatios respectively. [ ] ( 1 := 1 [ D t ] { }!, := 1 [ t=0 t ]!. (5.6 t=0 Here [ ] we have adapted the otatios due to Kuth (cf. [4] ad [9], where deotes the sigless Stirlig ubers of the first id, i.e., the uber of perutatios { of } obects havig cycles. Now, taig φ r (t = t r we have r φ r (0 =!, ad we see that (3. yields the forula r r { }! f ( r (0 = f ( (1. (5.7 0 =0 This forula iplies several iterestig special idetities. 9

10 [ D t ] t=1 = [ D (t + 1 ] t=0 (1 Taig f(t = e t, we get 1 r r { r e! = 0 =0 } = ω(r. (5.8 This is the well-ow forula of Dobisi for the Bell uber ω(r. ( Choosig f(t = 1 + t + + t ( 1 we fid f ( (0 =! for, ad f ( (0 = 0 for >, ad oreover, ( [( d (1 + t + + t dt =! t=1 + Thus (5.7 gives r = =0 r =0 ( + 1 ( { + 1 r! ( ] =! ( }. (5.9 This is the classical forula for arithetic progressio of higher order. (3 Taig f(t = 0 (tx = (1 tx 1 with tx < 1, we fid f ( (0 =!x ad f ( (1 =!x (1 x 1. Thus (5.7 yields r x = 0 r =0 { r! } x (1 x 1 ( x < 1. (5.10 This is Euler s forula for the arithetic-geoetric [ series. ] (4 Tae f(t = t so that f ( (0 = ( 1!. We have to copute f ( (1. By (5.6, it is easily foud that [ = (t + 1 t=0 D t 1] ( [D + t=0 1 1 t 1] t=0 [ ] [ = ( 1 1 1! + ( 1 1 ( 1! 1 ( [ ] [ ] =! ( ( ] Thus (5.7 gives [ ( 1 r =1 ] = r =1 { r! } ( ( 1 1 [ 1 This ay be copared with the ow idetity ] + ( 1 [ 1 1 (5.11 ]. 10

11 ( r = =1 r =1 { r! } (. (5.1 which is also obtaied fro (5.7 by taig f(t = (1 + t. (5 Choosig f(t = t := t(t + 1 (t + 1 ( 1 is arbitrarily fixed, we have ] t = Hece, f ( (0 =! [ f ( (1 = 1 = 1 f(t = [ 1 (! 1!!!! Therefore, (5.7 gives [ r 1 ] = r =1 1 ] ad fro (4 [D t ] 1 t=1 ( 1 1 1!! ( ( 1 1 [ 1!! ( { 1 r! 1 ( 1 t. 1 ] + ( 1 [ 1 1 } ( ( 1 1 [ 1 ]. ] + ( 1 [ 1 1 (6 Tae f(t = t(t a 1, the Abel polyoial with 1, so that f ( (0 = ( 1 1 ( a ad ]. f ( (1 = D [ t(t a 1] t=1 Thus (5.7 yields 1 = [ t( 1 (t a 1] t=1 + [ ( 1 1 (t a ] t=1 = ( 1 1 (1 a 1 [( + (1 a] = (1 a(1 a 1. r ( 1! ( 1 1 ( a = r =0 { r (1 a } (1 a 1. Exaple 4 Let α R. We have ( t + α = ( t + α ( 0. Thus, taig f(t = ( t+α we have f(0 = ( α. Cosequetly, (3.3 yields the idetity 11

12 1 =0 ( 1 4 ( + 1 ( ( α 1 [( α + 1/ = Exaple 5 For f(t = t ( 1 we have f(0 =! (3.3-(3.5, ad (3.7 give four idetities as follows 1 =0 =1 { ( ] α. (5.13 }, so that forulas ( 1 ( { } ( 1! 1 =, ( { } ( 1!H = ( 1, (5.15 { } { ( 1 1 if = ( 1!H 1 = 0 if >. (5.16 { } ( 1!H(, r = ( 1 r+1 (r 1 (5.17 = =1 Exaple 6 Taig f(t = ( +t, ( > 1, we fid ( + t = d dt Cosequetly, we have ( + t! ( 1 + t t t. f ( 1 = = ( 1! ( 1 ( (H 1 H 1 (H 0 = 0. Thus, usig (3.4 we get ( ( 1 ( 1 1 H = (H 1 H 1 (H 0 = 0. (5.18 =1 Exaple 7 Tae f(t = 1/(t + with. We have f(0 = ( 1!( 1! ( +! = ( 1 ( 1 +. Cosequetly forulas (3.4, (3.5, (3.7, ad (3.3 ca be used to obtai four coverget series sus as follows. 1

13 ( 1 + H = 1 ( ( 1 + H(, r = 1 ( ( 1, ( H + 1 = 1, (5.0 ( 1 r+ (r 1 (5.1 1 C 4 = + 1. (5. I particular, for = we see that (5.0, (5.1, ad (5. yield the sus 1 H ( + 1( + ( + 3 = 1 8, (5.3 H(, r = 1 (r 0 (5.4 ( + 1( + 1 ( 4 ( + 1( + ( + 3 = 1 5. (5.5 1 Exaple 8 As ay be observed, the case r = 0 of (3.6 gives the followig pair of idetities for f(t = t ad f(t = ( α+t (α R respectively. ( 1 (! { } ( 1 =, (5.6! =0 ( ( ( ( 1 α α 1 =. (5.7 4 =0 I particular, (5.7 with α = iplies =0 ( ( 1 4 ( = ( 1 ( =. (5.8 This idetity appears i Sofo [10, p.]. Surely, other idetities of siilar types ay be obtaied fro (3.6 for saller r s. Acowledgets We wish to tha the referees for their helpful coets ad suggestios. 13

14 Refereces [1] R. Mulli ad G.-C. Rota, O the foudatios of cobiatorial theory: III. Theory of bioial eueratio, i: Graph Theory ad its Applicatios, B. Harris (ed., Acadeic Press, New Yor ad Lodo, 1970, [] T. X. He, L. C. Hsu, ad P. J.-S. Shiue, Sybolizatio of geeratig fuctios, a applicatio of Mulli-Rota s theory of bioial eueratio, Cop. & Math. with Applicatios, 54 (007, [3] L. Cotet, Advaced Cobiatorics, the art of fiite ad ifiite expasios, Revised ad elarged editio. D. Reidel Publishig Co., Dordrecht, [4] H. S. Wilf, Geeratigfuctioology, Acadeic Press, New Yor, [5] L. B. W. Jolley, Suatio of series, d Revised Editio, Dover Publicatios, New Yor, [6] J. M. Satyer, A Stirlig lie sequece of ratioal ubers. Discrete Math. 171 (1997, o. 1-3, [7] Ch. Jorda, Calculus of Fiite Differeces, Chelsea Publishig Co., New Yor, [8] W. Magus, F. Oberhettiger, R. P. Soi, Forulas ad Theores for the Special Fuctios of Matheatical Physics, 3rd editio, Spriger-Verlag, Heidelberg, New Yor, [9] D. E. Kuth, Two otes o otatio, Aer. Math. Mothly 99 (199, o. 5, [10] A. Sofo, Coputatioal Techiques for the Suatio of Series, Kluwer Acadeic/Pleu Publishers, New Yor,