OVERVIEW OF THE COMBINATORICS FUNCTION TECHNIQUE

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1 OVERVIEW OF THE COMBINATORICS FUNCTION TECHNIQUE Alai J. Phaes Depatet of Physics, Medel Hall, Villaova Uivesity, Villaova, Pesylvaia, , USA, Hek F. Aoldus Depatet of Physics & Astooy, Mississippi State Uivesity, P.O. Dawe 567, Mississippi State, Mississippi, , USA, Abstact This pape suaizes the cobiatoics fuctio techique leadig to the solutio of a ultidiesioal liea ecuece elatio subject to a set of iitial coditios. Iteatioal Joual of Theoetical Physics, Goup Theoy ad Noliea Optics 7 () -3 Repited i: Thoas F. Geoge ad Hek F. Aoldus, editos, Theoetical Physics, pat, Hoizos i Wold Physics,, Vol. 43 (Nova Sciece Publishes, Hauppauge, New Yok, ) Pat 5, pages 9-4

2 . Multidiesioal liea ecuece elatios A ultidiesioal liea ecuece elatio (o patial diffeece equatio) ay be witte i the fo []: N B(M) = fa k (M)B(M Ak) ; M R. () The quatity B( M ) depeds o the set of vaiables {x, x,..., x } with which oe associates a poit, M, i a -diesioal Euclidea space havig fo coodiates the sae set of ubes. The positio vecto of that poit is deoted M, ad the ecuece elatio is valid fo all poits belogig to egio R. Oe also efes to B( M ) as the quatity B evaluated at poit M. Equatio () is a (N + )-te liea ecuece elatio, expessig that B( M ) is liealy elated to N othe values of B evaluated at poits with positio vectos { M A k ; k =,..., N}. Coefficiets fa k (M) ae labeled by the coespodig shifts A k ad ay also deped o the evaluatio poit M. Whe these coefficiets ae idepedet of the evaluatio poit, the liea ecuece elatio has costat coefficiets. While Eq. () states that the ecuece is valid fo poits M i egio R, i geeal the values of B i that egio ae ot uiquely deteied uless its values at soe othe poits ae specified. Assue that the values of B ae kow at bouday poits, J l, coespodig to positio vectos J l, accodig to: B (Jl) = λl ; l =,,... () The bouday poits fo a bouday egio R b = { J l ; l =,,... } ad the λ l s ae called iitial values. Coside the set of poits R J which ca be eached followig ay ube of displaceets ade of the eleets of set A = { A k ; k =,..., N}, ad leavig ay give bouday poit, without ecouteig ay of the othe bouday poits. A solutio to Eq. () satisfyig the bouday coditios () exists ad is uique, if ad oly if R R J. If thee is a bouday poit which caot be liked to ay of the poits i R without ecouteig aothe bouday poit, the the solutio does ot deped o the iitial value associated with this bouday poit. If egio R b does ot cotai such bouday poits, it is the called a iial bouday. Fo all pactical puposes, we will assue that R = R J ad that R b is a iial bouday. Thus the solutio of Eq. () satisfyig the iitial value coditios () is a liea cobiatio of all the iitial values listed i Eq. (). We shall wite this as []:

3 3 λ B (M) l C(Jl, M). (3) = l C(J, M The quatity l ) is called the cobiatoics fuctio. Its costuctio is based o all possible paths leavig the bouday poit J l ad eachig the evaluatio poit M by successive displaceets belogig to set A, while avoidig all othe bouday poits. The costuctio of a cobiatoics fuctio is peseted i the followig paagaph. A give path leavig J l ad eachig M is idetified by two labels, ω ad q. The foe label is the ube of displaceets i this path, ad the latte is used to distiguish the vaious discete paths havig the sae ube of q displaceets ω. Coside a give path (ωq) with displaceets δ q, δ q,..., δ i, q..., δ ω, ade i this ode, leavig J l ad eachig M. Let S i be the poit o this path eached afte the i th q displaceet, δ i. The positio vecto of that poit is: S i = Jl + δ i q j ; i =,,..., ω ; S o = Jl ; Sω = M. (4a) I this equatio we have itoduced fo coveiece the il displaceet, q = δ ; q. (4b) O the (ωq)-path, poit S i is eached followig displaceets δ q, δ,..., δ q i. With this poit oe associates the quatity f q (Si), ad with the (ωq)-path oe δi associates the poduct q ω Fω (J, M) = f q (S δ i ) i i= l, (5a) whee the coefficiet associated with the il displaceet is uity: f q (S) = f(s) = δ The cobiatoics fuctio is the give by:, S. (5b) q

4 4 ω q C(Jl, M) = Fω (Jl, M) = f q (S) δ. (6) ω, q ω, q i= By addig to the ight-had side of Eq. () a te, I( M ), which ay deped o the evaluatio poit, the ecuece elatio becoes ihoogeeous, N B(M) = I(M) + fak (M) B(M Ak) ; M R (7) The solutio of Eq. (7) satisfyig the bouday coditios of Eq. () is give by []: B(M) = λ C(J, M) + I(L) C(L, M). (8) l l l L R L, The cobiatoics fuctio C( M ) is costucted based o all possible paths (ade of displaceets belogig to set A) coectig poit L ad the evaluatio poit M ad avoidig the bouday poits. Equivaletly, oe ay also view the solutio of the ihoogeeous equatio as the su of two tes: the solutio of the hoogeeous equatio ad the paticula solutio of the ihoogeeous equatio coespodig to all the iitial values λ i =.. Oe-diesioal two-te ecuece elatios A. Geeal ethod With appopiate scalig, a two-te ecuece o a fuctio u(x) subject to the iitial value coditio, u(x o ) = λ, (9a) ay be witte as: u(x) = p(x) u(x ) + q(x), (9b) whee p(x) ad q(x) ae kow fuctios. The solutio of Eq. (9) is well kow [3] ad ay be obtaied by the ethod of vaiatio of paaetes, which elies o the kowledge of the solutio u (x) of the hoogeeous equatio, ad the seachig fo v(x) such that u(x) = v(x) u (x). A special solutio ay also be obtaied as a ascedig cotiued factio fo x > x o [3]. The cobiatoics

5 5 fuctio techique ay also be used to ecove the sae esult [4]. Fo defiiteess, let us assue that x > x o. Set A cotais oly oe eleet which is the displaceet by oe uit i the diectio of the positive x-axis. The fuctio associated with this displaceet is f (x) = p(x), ad the ihoogeeous te is I(x) = q(x). Regio R fo which the solutio is obtaied uiquely i tes of λ is the oe cotaiig the poits with coodiate x such that x x o is a positive itege, say. Thee exists oly oe path coectig ay two poits of coodiates y ad x fo which x y = is a positive itege. Thus the oediesioal cobiatoics fuctio associated with this path is C (y, x) = p(y + j), C(y, y) =, () ad the solutio of Eq. (9) follows: o, x) + q(xo + j) C(xo + j, x) u (x) = λc(x. () The ascedig cotiued factio solutio is also obtaied as a special case of the cobiatoics fuctio techique, whe eplacig Eq. (9b) by: u(x + ) q(x + ) u(x) = + ; x > x o. () p(x + ) p(x + ) Set A has oly oe eleet coespodig to the uit displaceet i the diectio of the egative x-axis, ad the fuctio associated with this displaceet is f (x) = /p(x + ). The ihoogeeous te is I(x) = q(x + )/p(x + ). A iial bouday egio associated with egio R ust cotai oe poit whose coodiate y satisfyig the iequality y > x > x o with the costait that x y = is a positive itege. The cobiatoics fuctio is deoted i this case as C*(y, x) to distiguish it fo the oe above, ad it is give by: * C (y, x) = ; C * (y, y) =. (3) p(x + j)

6 6 The geeal solutio of Eq. () with the bouday poit beig at a abitay locatio (y > x > x o ) the follows: + + = * q(x k ) (x) u(y)c (y, x) + * C (x + k, x p(x + k + ) ) u. (4) The ascedig cotiued factio solutio give by Mile-Thoso [3,5] is a special case of Eq. (4) whee the bouday poit is at ifiity ad u( ) =. The Eq. (4) is equivalet to: q(x + 4) +... q(x + 3) + p(x + 4) q(x + ) + p(x + 3) q(x + ) + p(x + ) u(x) =. (5) p(x + ) B. Beouilli ad Eule polyoials Let B (x) ad E (x) be the Beouilli ad Eule polyoials, espectively. They ae defied though the followig geeatig fuctios [6]: xt t e t = B (x) t e! =, t < π ; (6) xt e t = E (x) t e +! =, t < π. (7) They also satisfy the two-te ecuece elatios: + = + B + (x ) B (x) + ( + ) x ; (8) E (x) = E (x) + x. (9) We choose the bouday poit abitaily at x o such that x o = x [x], ()

7 7 whee [x] idicates the itege pat of x, which i tu shows that x o is i the age x o <. The iitial values fo ecuece elatios (8) ad (9) ae B + (x o ) ad E (x o ), espectively. I the Beouilli case, we have p(x + ) = ad q(x + ) = ( + ) x, ad Eq. () iplies [4]: [x] B (x) B (x ) ( k x ) o + o = + +. () + I the Eule case, p(x + ) =, q(x + ) = x ad Eq. () iplies [4]: [x] [x] [ x] k E (x) ( ) E (x ) ( ) (k x ) + o + o =. () These esults hold fo [x] with a positive itege. If x is a itege, say, the x o =, x = [x] =, ad Eqs. () ad () epoduce the kow esults [6] fo the su ad alteate su of cosecutive iteges which ae aised to the th powe. 3. Thee-te oe-diesioal ecuece elatios A. Legede polyoials The Legede polyoials P (z) satisfy the thee-te ecuece elatio [6]: (z) = z P (z) P (z),, (3a) P with the iitial values: P (z) =, P (z) =. (3b) A slight geealizatio of this poble is obtaied by addig a ihoogeeous te I(; z) to the ight-had side of Eq. (3a) keepig the sae iitial values [5]: B (z) = z B (z) B (z) + I(; z),. (4) Set A cotais two displaceet vectos i the diectio of the positive -axis ad of agitude ad, espectively. The fuctios associated with these displaceets ae:

8 8 f() = z, f() =. (5) I tes of the cobiatoics fuctios associated with this poble, the solutio is give as: (z) = C(, ) + I(j; z) C(j, ) B. (6) The cobiatoics fuctio C(,) is the Legede polyoial P (z). To coplete the costuctio of the geeal solutio, oe has to copute C(j,) fo > j. The esult is [5]: [( j) / ] C (j, ) = ( p j p ) z β, (7a) p= whee [q] efes agai to the itege pat of q, ad β depeds o j, ad p accodig to: β = j j! Γ( +! Γ(j + ) ) p p k ( )! (k + j )! k!(p k)! ( p + k)! (j )! Γ( + p + k) Γ(k + j ). (7b) Γ( + ) Γ(k + j ) Oe ay check that, fo j =, thee is oly oe o-zeo te i the k- suatio, the oe fo which k =. This, i tu, allows oe to exted the validity of Eq. (7) to j =, ad ecove the expessio of the Legede polyoial as C(j,), evaluated at j =. B. Fiboacci-like ecuece elatio Coside the Fiboacci-like ecuece elatio [7], B = a B + b B p, p, (8a) subject to the iitial values:

9 9 B, l =,,..., p -, B = λ. (8b) l p = λ l The Fiboacci ubes ae the solutios of Eqs. (8) whe a = b =, p =, λ = ad λ =. The stadad ethod of solvig Eqs. (8) would be to seach fo paticula solutios of the fo: B = R, (9) which i tu equies R to be a oot of the chaacteistic equatio R p a R p b =. (3) Let R k be oe of the p oots with idex k vayig fo to p. The the solutio of Eq. (8a) is a liea cobiatio of these p special solutios, p B = L R, (3) k k with the L k s satisfyig the iitial value coditios (8b). The cobiatoics fuctio techique povides a siple way of obtaiig the aalytical solutio fo ay p. Hee set A is ade of two displaceet vectos alog the positive - axis of agitudes ad p, with the associated fuctios f () = a ad f p () = b. Coside all paths eachig the evaluatio poit of coodiate, ad leavig ay of the bouday poits of coodiate j (j =,,..., p ) while avoidig all othe bouday poits. Thus, the fist displaceet δ fo all of these paths is esticted to be equal to p fo as log as j, ad the cobiatoics fuctio C( j, ) follows as [7]: j [ p ] j kp k+ δ j k(p ) j C ( j, ) = a b, (3) k whee the quatity i backets epesets a bioial coefficiet. The solutio to Eq. (8a) satisfyig the iitial value coditios (8b) is: p = λ j C( j, ) B. (33)

10 I this ae, we avoided calculatig the oots of the chaacteistic equatio ad the p coefficiets L k i tes of the p iitial values λ j. O the othe had, the equivalece betwee the two ethods yields a su ule valid fo, p ad fo ay values of a ad b, aely [7], [ p ] a kp b k k(p ) = k p R + k p k a + bpr. (34) 4. The Schödige equatio with a powe-type potetial Coside a paticle of ass i a cetal potetial V() = K N, (35) whee N is a positive itege. The adial pat R() of the wave fuctio descibig the statioay state of the paticle with eegy E is witte as R() = U()/, (36) with U() the well-behaved solutio of the adial Schödige equatio: d U dρ l l( l + ) N + ρ ρ t U l =. (37) Hee, l is the obital quatu ube, ad t ad ρ ae diesioless eegy ad adial vaiable paaetes: h t = N N+ K N+ E ; K ρ = h N+. (38) The behavio of U l fo lage values of ρ is the sae fo all values of l. This behavio is that of the solutio of Eq. (37) with l =, aely, d Uo dρ ( ρ N t) U o =. (39)

11 Fo the liea potetial (N = ), the well-behaved solutio of Eq. (39) is the Aiy fuctio [6], Ai(ρ t). Sice R() ust be a well-behaved fuctio i the liit as appoaches zeo, oe should also equie that Ai(ρ t) = as ρ appoaches zeo; thus: Ai( t) =. (4) The oots of this equatio povide the s-state ( l = ) eegy eigevalues fo the liea potetial poble (N = ). Hee we ited to develop a eegy eigevalue equatio valid fo positive itege values of N which educes to Eq. (4) whe settig N = ad l =. By factoig out the o-sigula behavio of U l ea the oigi, we seach fo a seies solutio of the fo: l+ Ul = ρ bρ. (4) = Substitutig this seies ito Eq. (37) yields the thee-te ecuece elatio: ( + l + ) b = t b + b N, >, (4a) subject to the iitial value coditios: b = fo >, b. (4b) We ited to expess b i tes of highe ode tes with the bouday at poits o the -axis with coodiates { + M + j ; j =,,..., N + }, with M beig at this poit a uspecified positive itege. This is why we use a equivalet fo of Eq. (4a), b ( + N + ) ( + N + + 3) b+ N+ + t b+ N = l. (43) Set A fo this ecuece elatio cosists of displaceets i the diectio of the egative -axis with agitudes N + ad N, ad thei associated fuctios ae: F (N + ) () = ( + N + )( + N + l + 3), f N () = t. (44)

12 With C( + M + j, ) desigatig the cobiatoics fuctio, the solutio of Eq. (43) i tes of the coefficiets b + M + j (j =,,..., N) is: N + = b + M + j C( + M + j, ) b. (45) This expessio holds fo ay value of. Futheoe, Eq. (4a) with the bouday value coditios (4b) iplies + N = ( l ) b = t b + b. (46) Cobiig Eqs. (45) ad (46) yields N + = b+ M+ j C( + M + j, ) b. (47) Next, we coside a double seies expasio of U o (ρ) of the fo [9] i o ( ρ) = ρ ( t) A(,i) ρ b ( =, t) = i= = U l, (48) which exists fo N, ad its expasio coefficiets A(, i) satisfy the ecuece elatio: ( ) A(, i + ) A(, i) A( N, i + ) =, (49a) with the iitial value coditios: A(, i) = fo >, A(, ). (49b) Sice the asyptotic behavio of Ul ( ρ) is the sae as that of U o (ρ) fo all values of l, i the liit as becoes ifiite, coefficiet b ( l, t) should be popotioal to b ( l =, t): i b (, t) b ( l =, t) = ( t) A(,i) i= l. (5) Usig Eq. (5) i takig the liit of Eq. (47) as M appoaches ifiity yields:

13 3 b H (t) = l N N+ i = ( t) li A( + M + j,i) C( + M + j,) = M i =. (5) This ifiite ode polyoial i t is the geealizatio of the oe obtaied i Ref. fo the liea potetial (N = ) ad holds fo all positive values of N except N =. Excludig the haoic potetial (N = ), the oots of H l N (t) ae the eegy eigevalues of the positive powe potetial poble. With a appopiate choice of A(, ), H l N (t) educes to Ai( t) fo N = ad l =. Ufotuately, we have bee uable to fid a closed fo expessio fo the expasio coefficiets fo abitay N ad l. I the liea potetial case (N = ), we wee able to copute i closed fo the fist few lowe ode tes fo l =. 5. Suay Multidiesioal liea ecuece elatios ca be foally solved i tes of the cobiatoics fuctios. A cobiatoics fuctio depeds o a bouday poit ad a evaluatio poit. Bouday poits ae deteied by the iitial value coditios ad the evaluatio poits ae those at which oe would like to deteie the value of the ukow i tes of the iitial values. The costuctio of a give cobiatoics fuctio is based o all possible paths coectig a bouday poit to a evaluatio poit. These paths ae ade of discete displaceets. The agitudes ad diectios of these displaceets ae eadily idetified fo the ecuece elatio. The ube of such displaceets is oe less tha the ube of tes i the ecuece elatio, ad a oe-to-oe coespodece is established betwee these displaceets ad the coefficiets appeaig i the ecuece elatio. A path coectig a bouday poit to a evaluatio poit is the ade of a give sequece of displaceets. With such a path oe associates the poduct of the coespodig coefficiets evaluated at the successive iteediate poits ecouteed alog the path. The costuctio of the cobiatoics fuctio fo a give set of bouday ad evaluatio poits follows as the su of such poducts, coespodig to all possible paths leavig the bouday poit ad eachig the evaluatio poit, while avoidig all othe bouday poits. Fo a few cases we have show how kow esults usig othe ethods ae ecoveed usig the cobiatoics fuctio techique, while povidig soe atual geealizatios. Not peseted i this

14 4 aticle is the use of the cobiatoics fuctio techique to iclude the solutio of liealy coupled ecuece elatios []. Such elatios ivolve a set of ultidiesioal fuctios B i (M). The value of a give fuctio Bj at the evaluatio poit M is liealy elated ot oly to its values at othe evaluatio poits but also to the values of the eaiig fuctios evaluated at shifted aguets. A geeal ethod fo decouplig these equatios has bee peseted i Ref.. I the case of liealy coupled elatios with costat coefficiets [], the decouplig is uch siple to achieve tha i the geeal case. Applicatios of this decouplig has bee exteely useful i the study of Isig odels [3, 4]. This techique was also istuetal i developig a coputatioal ethod fo the exact study of low tepeatue adsoptio pattes o cystal sufaces of fiite width ad ifiite legth [5]. We foud that the cystallizatio pattes o sei-ifiite sufaces without peiodic boudaies have chaacteistics which fit exact aalytic expessios as a fuctio of the width of the suface. Refeeces [] A. J. Phaes ad R. J. Meie, J., J. Math. Phys., (98). [] A. J. Phaes, J. Math. Phys., 3 (98). [3] L. M. Mile-Thoso, The Calculus of Fiite Diffeeces (St. Mati s, New Yok, 95). [4] A. J. Phaes, J. Math. Phys. 8, 838 (977). [5] L. M. Thoso, Poc. Edibugh Math. Soc., 3 (933). [6] See, fo exaple, Hadbook of Matheatical Fuctios, edited by Abaowitz ad I. A. Stegu (Dove, New Yok, 97). [7] A. J. Phaes, Fiboacci Quat., 9 (984). [8] B. J. Haigto ad A. Yildiz, Phys. Rev. Lett. 34, 68 (975). [9] A. J. Phaes, Villaova pepit # VU-TH3-. [] A. J. Phaes, J. Math. Phys. 9, 39 (978). [] A. J. Phaes, Poc. of the Fifth IMACS Iteatioal Syposiu, Advaces i Copute Methods fo Patial Diffeetial Equatios, Vol. V, 53 (984). [] A. J. Phaes, J. Math. Phys. 5, 69 (984). [3] A. J. Phaes, J. Math. Phys. 5, 756 (984). [4] See, fo exaple, J. L. Hock ad R. B. McQuista, J. Che. Phys. 83, 366 (985); D. Walikaie ad R. B. McQuista, J. Math. Phys. 6, 85 (985);

15 5 R. B. McQuista ad J. L. Hock, J. Math. Phys. 7, 599 (986); A. J. Phaes ad F. J. Wudelich, J. Math. Phys. 7, 99 (986); Phys. Lett. A, 347 (987); Il Nuovo Ci. B, 653 (988); Phys. Lett. A 3, 385 (988). [5] A. J. Phaes, F. J. Wudelich, J. D. Culey ad D. W. Gubie, J., J. Phys. A 6, 6847 (993); A. J. Phaes ad F. J. Wudelich, Phys. Rev. E 5, 36 (995); Phys. Lettes A 6, 336 (997); A. J. Phaes, F. J. Wudelich, J. P. Mati, P. M. Bus ad G. K. Duda, Phys. Rev. E 56, 447 (997); A. J. Phaes ad F. J. Wudelich, Suf. Sci. 45, (999).