On the Rodrigues Formula Approach. to Operator Factorization

Transcription

1 Iteratioal Mathematical Forum, Vol. 7, 0, o. 47, O the Rorigues Formula Approach to Operator Factorizatio W. Robi Egieerig Mathematics Group Eiburgh Napier Uiversity 0 Colito Roa, Eiburgh, EH0 5DT, UK b.robi@apier.ac.uk Abstract I this paper, we erive geeral formulae that reprouce well-kow istaces of recurrece relatios for the classical orthogoal polyomials as special cases. These recurrece relatios are erive, usig oly elemetary mathematics, irectly from the geeral Rorigues formula for the classical orthogoal polyomials a first-priciples erivatio a represet a uifie presetatio of various approaches to the exact solutio of a importat class of seco-orer liear oriary ifferetial equatios. Whe re-expresse i laer-operator form, the recurrece relatios are see to represet to a basic evelopmet of the work of Jafarizaeh a Fakhri [5] a allow a Schröiger operator factorizatio of the efiig equatio of the classical orthogoal polyomials, as well as a operatioal formula for the solutio of this efiig equatio. The ietity betwee the Rorigues formula a the operatioal formula is etermie a staar examples ivolvig the applicatio of the laer-operator approach presete. The relatioship with previous work is iscusse. Mathematics Subject Classificatio: 33C45; 4C05 Keywors: classical orthogoal polyomials; laer-operators; operatioal formula. Rorigues Formula Solutios to Seco-Orer Differetial Equatios I this paper we cosier the seco-orer liear oriary ifferetial equatio

2 334 W. Robi p(z y (z q(zy (z λ y (z 0 ( = where 0 is a o-egative iteger, the ashes eote ifferetiatio with respect to the fuctio argumet z, λ is iepeet of z, a p p(z = z p (0z p(0 a q (z = q z q(0 ( that is, p(z is a quaratic fuctio, q(z a liear fuctio of z. Equatio ( has kow (classical orthogoal polyomial solutios, with ormalisig factors y K (z = [w(zp (z] (3 K w(z with respect to the weight fuctio w (z i the iterval (a, b which iterval ee ot be fiite provie that [3] ( p q = λ (4 Give a oriary ifferetial equatio (., the weightig fuctio w(z is etermie by a first-orer oriary ifferetial equatio, a Pearso equatio [] [w(zp k (z] = [q(x kp (z][w(zp k (x] (5 with k 0 a o-egative iteger. As p(z a q(z are give, (5 may be solve for w(z a we have the (up to a ormalisatio factor a solutio (3 of (, provie that p(z, q(z a assume throughout. λ satisfy the itegratio coitio (4 (which we Now, aother well-kow approach to the solutio of (, is to (effectively factorize the seco-orer liear ifferetial operator o the left-ha sie of ( calle here the Schröiger factorizatio approach to the solutio of (. Sice the solutio of ( is uique, it is apparet that the two solutio

3 Rorigues formula approach to operator factorizatio 335 processes for ( the Rorigues formula solutio a the factorizatio approach must yiel the same aswers uer the same set of circumstaces, a the questio arises as to the exact coectio betwee the Rorigues formula solutio to ( a the Schröiger factorizatio approach to (. It is this questio that we aress, below, i the boy of the paper. There is a cosierable volume of literature o the Rorigues formula a factorizatio approaches to equatio ( a its solutio. For the purpose of compariso, however, we restrict the iscussio of this literature to a few of the most pertiet refereces. Of particular iterest here, by way of compariso with the results presete below, is the work of Erelyi et al [3], Jafarizaeh a Fakhri [5], Lorete [8], Kaufma [6], Nikiforov a Uvarov [], Va Iseghem [3] a Yaez, Dehesa a Nikiforov [4], which we iscuss i etail i sectio 5, (Of course referece [3] has bee a staar for may years. The approach to the Schröiger factorizatio that we evelop below is base o the emostratio, i sectio, of recurrece relatios for the Rorigues formula (3. The emostratios i sectio are base o techiques goig well-back [], [4] that epe o the mathematical structure of the Rorigues' formula (3. The laer-operator formalism that emerges from the aalysis of the mathematical structure of the Rorigues' formula (3, presete i sectio 3, is first cousi to that evelope by Jafarizaeh a Fakhri [5], who, however, obtai their laeroperators by factorizig (the equivalet of ( irectly. As a evelopmet, a extra elemet is here extracte a a assumptio elimiate from the Jafarizaeh a Fakhri formalism a a operatioal ietity establishe betwee the

4 336 W. Robi Rorigues formula solutio a the laer-operator solutio to (. Examples of the laer-operators metho are presete i sectio 4 a we rou-off our presetatio with a brief iscussio a ackowlegemet of the work of previous authors, a some further closig remarks, i sectio 5.. The Recurrece Relatios for the Rorigues Formula The basic relatios that lik the factorizatio methoology with the Rorigues formula solutio to ( [(3] are recurrece relatios for (3. I this sectio the erivatios of ifferetial recurrece relatios a a three term recurrece relatio for the Rorigues formula solutio, (3, to equatio ( are outlie. I the ext sectio we show how the ifferetial recurrece relatios for (3 ca be use to set-up the Schröiger factorizatio approach to solvig (. The techique that we aopt i the erivatios below is to erive a maipulate, through elimiatio, equatios ivolvig erivatives of w(zp (z. The philosophy motivatig this approach is obvious o examiig the structure of (3. We require five equatios i total, of which the first is the Pearso equatio (5. The remaiig four equatios are obtaie by ifferetiatig (5 a by the applicatio of Leibitz rule for ifferetiatig a prouct. So, from (5 we fi that ( k = k ] = (q p ] (q p ] (6 while, from the irect applicatio of Leibitz rule to p(z[w(zp (z], we get

5 Rorigues formula approach to operator factorizatio 337 ] = p ] ( p ( ] p (7 The fourth a fifth equatios are obtaie i a maer similar to (6 a (7. From (5 we fi that ( k = 3 ] = (q ( p ] ( (q ( p (8 a, from the irect applicatio of Leibitz rule to p(z[w(zp (z], we get 3 ] = p ] ( p ( ( ] p (9 We write-out, first, the expressio for the raisig ifferetial recurrece relatio, the we preset the three term recurrece relatio a, fially, we write-out the expressio for the lowerig ifferetial recurrece relatio. ] ] ] So, elimiatig ] a ] from (5, (8 a (7, with (3 i mi, we get the raisig ifferetial recurrece relatio y (z (q ( p (q p (q ( p K p y (z = y (z (q p (q p K (Note that, from (3 a (5 with k = 0, we fi that [] (0

6 338 W. Robi p ] = K w[py (q p y ] Next, from (7, (8 a (9 we may elimiate ], 3 ] a ] to get, with (3 i mi, the three term recurrece relatio i the form q ( p (q ( p K y ( (z ( q ( p {(q ( p p (q p Q}K y (z = (q p {(q ( p p (q ( p [(q ( p p p q]}k y (z Fially, elimiatig y (z from (0 a ( we fi the lowerig ifferetial recurrece relatio y (z [(q ( p p p q] p y (q ( p {(q ( p = (z p (q ( p [(q ( p p p q]} K (q ( p K ( y (z ( 3. The Schröiger Factorizatio Metho Oe of the best-kow ways of proviig solutios to equatio ( is to itrouce laer operators [5], [6], [8]. Laer-operators are first-orer liear ifferetial operators that relate ifferet solutios of ( to oe aother via ifferetiatio a elemetary algebra. The big avatage of employig laer-

7 Rorigues formula approach to operator factorizatio 339 operators is that give ay oe of the laer-operator solutios, ay other solutio ca be obtaie simply by applyig the appropriate laer-operator a sufficiet umber of times. I aitio, it is possible to re-arrage the basic equatio, (, so that it may be represete i a particular factorize form that we have calle here the Schröiger factorize form or Schröiger factorizatio. Specifically, the laer operators come i matche pairs : a raisig laer operator, L L = p(z ( αz β (3 with α a β iepeet of z, a a lowerig laer operator, L L = p(z ( αz β (4 with α a β iepeet of z also. The basic property of the raisig laer operator, L, is that it relates successive solutios of ( through L y (z = γy (z (5 while the correspoig relatio for the lowerig laer operator, L, is with γ Ly (z = γy(z (6 a γ iepeet of z. Usig the laer operator represetatio, (3 a (4, we may reprouce the seco-orer liear ODE ( (or its equivalet by combiig (5 with (6 as either, L see immeiately that L or L L = γ γ. Iee if we efie E (7

8 340 W. Robi a L L y (z E y (z (8 = LL y(z = Ey(z (9 The relatioship betwee the Rorigues formula solutio a the laer operator approach to ( is, i the light of sectio, immeiate. If we rewrite (5 a (6 i full, usig (3 a (4 respectively, the we get a y (z (z ( αz β y (z = γy (z (0 p y (z (z ( αz β y (z = γy (z ( p Comparig (0 with (0, we ietify the coefficiets of (3 as ( = q p α (a a β (q ( p [q(0 p (0] = (b (q p ( q ( p K γ = (c (q p K Also, by comparig ( with (, we fi that a γ = α β p = [(q ( p p (0 p q(0] = (q ( p (3a (3b {(q ( p p (q ( p [(q ( p p p q]} K (q ( p K (3c Further to this, combiig the results from equatios (c a (3c, we get

9 Rorigues formula approach to operator factorizatio 34 = γ a expressio for E γ as E (q ( p { (q ( p p (q ( p [(q ( p p p q]} = 4(q ( p (4 Fially, we still have a three-term recurrece relatio betwee the ifferet levels of solutio to (, that is y (z (a z b y (z c y (z (5 where the coefficiets a, b a c, with a, b a c iepeet of z, ca be evaluate by a basic compariso betwee (5 a (. Iterestigly, the ormalizatio factors for the obtaie irectly from (5 [7]. y (z, the classical orthogoal polyomials, may be The laer operator represetatio gives us, the, a alterative solutio metho for the ODE (, as metioe above. I fact give y 0 (z, we may obtai all other solutios y (z,, from equatio (5, sice 0, 0. γ To obtai y 0 (z, we make use of the fact, from ( a (3, that y 0 (z is obtaie as a solutio to I other wors, y 0 (z 0 = (6 y 0 (z is a costat, which is well-kow [3, ] from the theory of classical orthogoal polyomials. This is, of course, oly the case if we ietify the equivalece betwee (8, say, a the origial equatio (. Iee, if we write-out (8 i full a compare the result with ( i the form p(z[p(z y (z q(zy (z λ y (z] 0 (7 =

10 34 W. Robi the we see that we require the cosistecy coitios a α p (z ( α α z ( β β q(z (8 p(z ( α z β ( αz β E p(z λ (9 Similarly, if we write-out (9 i full a compare the result with the ajuste equatio (7 ( the we see that we require the further cosistecy coitio p(z ( α z β ( αz β E p(z λ α (30 A straightforwar calculatio, a matter of substitutio a simplificatio, shows that the ietities (8 to (30 are iee satisfie. I Table, we preset the factorize form of the Legere, the Laguerre a the Hermite equatios, alog with their respective Rorigues formulae. By costructio, the laer operator represetatio gives ietical aswers to the Rorigues formula. Iee, by repeate applicatio of the raisig operator o y 0 (z, we get a so-calle operatioal formula for y (z, that is y (z = ( L k k y0(z γ k = 0 (3 By choosig (see [3] a [] y 0 (z, we must therefore expect to fi y (z = k = 0 ( γ k L k = K w(z [w(zp (z] (3 To cofirm this, we may procee by mathematical iuctio o. First, for = 0, we have y 0 (z i both cases, as K 0 by covetio (see [3] a [] agai. Next, we assume the result (3 is true for arbitrary > 0. Fially, we have

11 Rorigues formula approach to operator factorizatio 343 y (z ( L k k [( k Lk ] = γ ( γk L = γ k (33 k = 0 k = 0 or, by (3 a the iuctio assumptio y (z ( L = k k = [( γk Lk ] [w(zp (z] γ (34 k= 0 Kw(z However, from (3, ( a (0, we see that [( γ k Lk ] [w(zp (z] = [w(zp (z] (35 K w(z K w(z a, from (33 a (34, the assertio hols true. 4. Examples It is ot a ifficult task to apply the geeral formulae we have presete above to ay specific class of classical orthogoal polyomials. We cosier i etail the geeratio of the first few terms of the Jacobi polyomials, while presetig similar results for the Legere, Laguerre a Hermite polyomials i tabular form i Table. The Jacobi polyomials arise as solutios [3] of the Jacobi equatio ( z P [ β α ( α β z]p With the ormalizatio ecapsulate i ( α β P = 0 (36 K = (! (37 we obtai the recurrece relatio geeratig the Jacobi polyomials, via (36, (0, (, a (3, as

12 344 W. Robi P ( k (z (z α β = k = 0 (k ( α β k Choosig [3] ( β α ( α β k z P (k 0 (z (38 P0 (z = (39a we see from (38 that a P α β ( α β z (z = (39b P ( α β 3( α β 4z (z = ( α ( α β 3z α 3 β α(β β (39c 0 a so o a so forth. We recogise P (z, P 0 (z a P (z as the first three terms of the sequece of Jacobi polyomials geerate by (38 a (39a. Naturally, the Rorigues formula (3, with the appropriate choice of K, w (z a p (z evelops the ietical sequece of Jacobi polyomials also, as may be checke from the actual formula, that is, from [3] P ( α β α β (z = ( z ( z [( z ( z ] (40! I Table we summarize the above proceure for the Legere, Laguerre a Hermite polyomials. As basic ata for the costructio of Table, we have the followig. For the Legere polyomials we have the equatio a ormalizatio ( z P zp ( P = 0, K = (! (4 while for the Laguerre polyomials a Hermite polyomials we have, respectively

13 Rorigues formula approach to operator factorizatio 345 a zl ( zl L 0, K =! (4 = zh zh H = 0, K ( = (43 Fially, i all three cases, Legere, Laguerre a Hermite polyomials, we assume that y 0 (z =. Agai, the Rorigues formula (3, with the appropriate choice of K, w (z a p (z evelops the ietical sequece of Legere, Laguerre a Hermite polyomials also. This may be checke usig the explicit Rorigues formulae from Table. Of course, the aalysis presete here may be applie to the other examples of classical orthogoal polyomials also (see refereces [3] a [] for their etaile escriptio.the basic process, summarize by (0 a ( (essetially efies a geeric ifiite sequece for the geeral classical orthogoal polyomial solutios of equatio (. As the process is iterative, it may be implemete o a computer a as may or as few terms i the sequece of polyomials ture-out as is require. The same is equally true of the recurrece relatio (5, give y 0 (z a y (z. I this case we require the coefficiets a, b a c, but this is a straightforwar etermiatio as metioe alreay. Also, it is ot ifficult to show that α =α γ a, β =β γ b, γ =γ c (44 from which the coefficiets a, b a c may be fou by basic arithmetic. 5. Summary a Discussio We have erive laer-operator ifferetial recursive formulae a a

14 346 W. Robi associate operatioal formula for the solutios of (, by eucig the recurrece relatios (0, ( a ( irectly from the Rorigues formula (, usig the Pearso equatio (4 a elemetary mathematics oly. The laer-operators (3 a (4, followig from the recurrece relatios (0 a (, lea the to a Schröiger factorizatio for (. I Tables a, a short collectio of some of the results of this aalysis is presete. The erivatios i sectio, ispire (mostly by the methoology of refereces [] a [4], costitute a first-priciples euctio of the recurrece relatios (0, ( a ( for the classical orthogoal polyomials (3 a so for the Schröiger laer-operators (3 a (4 for equatio ( also. Note, however, that the presetatios of refereces [] a [4] are base o a ifferet startig formula tha (3 so that their erivatios iffer i etail from those of sectio, as o the specific form of their fial results. The recurrece formulae of sectio are well a wiely kow. However, their erivatio through the maipulatio of the geeral Rorigues formula (3 is ot of such a wie currecy a previous essays i this irectio iffer from the approach of sectio either i startig poit, as i refereces [] a [4], or else i scope of applicatio. So, o the oe ha, the recurrece relatios of sectio have bee obtaie from the Rorigues formula solutio of ( by Nikiforov a Uvarov [] a also, i a evelopmet of the work of Nikiforov a Uvarov [], by Yaez, Dehesa a Nikiforov [4]. However, Nikiforov a Uvarov [] a Yaez et al [4] o ot erive the recurrece relatios from the Rorigues formula (.3 irectly, but through a aalysis, via cotour itegratio, of a

15 Rorigues formula approach to operator factorizatio 347 cotour itegral represetatio of (3. Similarly, Erelyi et al [3] quote their equivalet to (0, ( a ( (with their (0 implicit i the iscussio. However, their erivatio iffers i startig poit (a i etails also. O the other ha, parallel to the metho evelope i sectio is that presete by Va Iseghem [3]. Va Iseghem oes maipulate (3 irectly, but erives equatios (6 a (7 oly, a so prouces solely the raisig ifferetial recurrece relatio, that is, equatio (0. (Va Iseghem [3] oes erive her equivalet of (, but agai from the viewpoit of Nikiforov a Uvarov []. I aitio to these iffereces betwee the work i sectio, a that of refereces [3], [], [3] a [4], we ote that, the coefficiets i the recurrece formulae of sectio are give i terms of the coefficiets of the origial secoorer liear ifferetial equatio, (, irectly, thus makig applyig these recurrece formulae, to ay particular istace of a ifferetial equatio, simpler. Noe of the above quote authors relate their recurrece relatios to the factorizatio of equatio (. So, we move o a cosier previous work associate with the evelopmet of laer-operators alog the lies of sectio 3. Agai, such formulae as (0, ( or (5 are well kow, but, oce more, essays i evelopig formulae such as (0, ( or (5 iffer from the approach of sectio 3 either i startig poit or i scope of applicatio. For example, Lorete [8] erives a Schröiger operator factorizatio for, a solutios to (orthogoal polyomials equatio (, through the applicatio of recurrece relatios erive from its Rorigues formula by Nikiforov a Uvarov [] a so (implicitly Lorete has a ifferet startig poit from that of sectio

16 348 W. Robi 3. Actually, Lorete [8] prouces a factorize form of a equatio relate to (. As a seco example, we cosier Kaufma [6], who also starte with laer operator relatios a proceee to costruct seco-orer oriary ifferetial equatios i the maer of Lorete. Kaufma, however, cosiere oly special cases of laer operators [6], which he cosiere as give. I the Rorigues formula proceure, laer operators are erive for the geeral Rorigues formula, (3 (or see Lorete [8] agai. Aother laer-operator methoology, a oe which is closest to that presete i sectio 3, is that of Jafarizaeh a Fakhri [5]. I fact, the preset work esup, as reporte i sectio 3, as a evelopmet of the factorizatio methoology of Jafarizaeh a Fakhri [5]. However, Jafarizaeh a Fakhri [5] start, ot with the Rorigues formula (3, but with the assume factorizatios (8 a (9 of equatio (. I aitio, Jafarizaeh a Fakhri [5] o ot assume the specific forms (0 a ( for the raisig a lowerig operatios but utilize, istea, the composite format of a ŷ (z (z ( αz β ŷ (z = ŷ (z (45 p ŷ (z (z ( αz β ŷ (z = Eŷ (z (46 p ±, β ± a, to etermie the ukow coefficiets α a E, require the further assumptio that E 0 = 0 [5]. The use, i sectio 3, of the irect approach of sectio γ prouces the factorize form E = γ, from which, as 0, we get γ0 = E 0 = 0 ietically, as a result. Further, the operatioal formula (3 epes o

17 Rorigues formula approach to operator factorizatio 349 the ifferet approach of sectio 3. Apparetly, by compariso, the relatioship betwee the approach of Jafarizaeh a Fakhri [5] a the preset approach, is etermie through provie we agree that (z = ŷ (z. y (z = ( k ŷ (z, γ (47 k = y0 0 = We rou-off our iscussio with a few geeral poits of iterest. First, we ote that the Schröiger factorizatios i aitio to beig well-kow i the theory of Sturm-Liouville eigevalue problems a itimately relate to the cocepts of supersymmetry a shape ivariace [5], are basic to the theory of special fuctios [0]. Iee the factorizatio approach to ( les itself to further evelopmets, especially where the task of likig geeratig fuctios a recurrece relatios is cocere. I particular, by makig use of recurrece relatios, we ca utilise existig methoology (see [9] a [] to prouce a multitue of geeratig fuctios, which are kow to be of great utility i applicatios. Fially, we ote that we have restricte ourselves to the basic classical orthogoal polyomials. The methoology for proucig raisig a lowerig operators a factorize forms ca be extee to a cosieratio of the associate polyomials a their efiig ifferetial equatio, although the formulae are somewhat more ivolve. Refereces [] Beale F.S., Aals of Math. Stats. (94 97.

18 350 W. Robi [] Chihara T. S., A Itrouctio to Orthogoal Polyomials (Goro a Breach 978. [3] Erelyi A., Magus W., Oberhettiger F. a Tricomi F. G., Higher Trasceetal Fuctios Volume II (McGraw-Hill 953. [4] Hilebrat E.H., Aals of Math. Stats. ( [5] Jafarizaeh M. A. a Fakhri H., Physics Letters A 30 (997 64; A. Phys. (N.Y. 6 ( [6] Kaufma B., J. Math. Phys. 7 ( [7] Law A. G. a Sle M. B., Proc. Am. math. Soc. 48 ( [8] Lorete M., J. Phys. A: Math. Ge. 34 ( [9] McBrie E. B., Obtaiig Geeratig Fuctios (Spriger-Verlag 97. [0] Miller W., Lie Theory a Special Fuctios (Acaemic Press 968. [] Nikiforov A. F. a Uvarov V. B., Special Fuctios of Mathematical Physics (Birkhauser 988. [] Truesell C., A Essay Towar a Uifie Theory of Special Fuctios (Priceto Uiversity Press 948. [3] Va Iseghem J., ScietificCommos (995 [4] Yaez R. J., Dehesa J. S. a Nikiforov A. F., J. Math Aal. Appl, 88 ( Equatio Equatio Form Factorize Form Rorigues' Formula Legere ( z P zp ( P = 0 [( z z][( z z]p = P ( P = [( z ]! Hermite H zh H = 0 z( H H ( = z z H = ( e [e ] Laguerre z L ( zl L = 0 ( z z(z L L = x z L = e [z e ]! Table. Importat Equatios, Factorize with their Rorigues' Formula [3]

19 Rorigues formula approach to operator factorizatio 35 Term Legere Recurrece Laguerre Recurrece Hermite Recurrece P (z = ( z (k z k = 0 k L (z zl k (k zl k = k= 0 k H (z = (H k (z zh k k= 0 0 z z z 3z z 4z 4z 3 3 5z 3z 3 z 9z 8z z z z 30z z 6z 7z 96z z 48z Table. Recurrece Relatios a the First Few Terms of their Sequeces Receive: April, 0