Discrete q-hermite polynomials: An elementary approach

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1 Discrete -Herite polyoials: A eleetary approach Joha Cigler Faultät für Matheati, Uiversität Wie Ui Wie Rossau, Osar-Morgester-Plat, 090 Wie johacigler@uivieacat Abstract We preset a siple approach to discrete Herite polyoials with special ephasis o aalogies with the classical case 0 Itroductio The so-called discrete Herite polyoials are aalogues of the classical Herite polyoials which geeralie ost of their eleetary properties The purpose of this paper is to ephasie these aalogies without recourse to sophisticated geeral theories Soe 35 years ago I stubled upo these polyoials (cf [3], [4]) as I studied soe idetities fro the poit of view of Rota s ubral calculus ([5]) I his thesis Peter Kirschehofer [] obtaied further properties ad deteried a easure o the real lie with respect to which they are orthogoal At that tie I was uaware that Al-Sala ad Carlit [] had itroduced these polyoials soe years earlier via geeratig fuctios Soe properties of these polyoials are collected i [3] withi the fraewor of the Asey- schee of hypergeoetric orthogoal polyoials ad its aalogue They are also dealt with i a eleetary way i [8], [9] ad [4] Here I give a itroductio to these polyoials i the spirit of y papers [3], [4] I shall derive recurrece relatios, geeratig fuctios, Rodrigues type forulae together with aalogues of Burchall s ad Nielse s forula ad soe easures with respect to which the polyoials are orthogoal Fially a curious coectio with taget ad Euler ubers is etioed Whereas ost results are well ow or hidde i ore geeral theories the approach sees to be ovel Bacgroud aterial about Herite polyoials I this sectio I state soe well-ow results about the classical (probabilists ) Herite polyoials (cf [], [5]) which have beautiful geeraliatios for discrete Herite polyoials The ai iterest is i H ( ) H(,) but it is coveiet to state soe results ore geerally for bivariate Herite polyoials For s 0 the bivariate Herite polyoials 0 () H(, s) ( ) ()!! s

2 are oic polyoials which satisfy ad are orthogoal with respect to the liear fuctioal defied by They satisfy the recurrece with iitial values H (, s) ad 0 Their geeratig fuctio is d H ( s, ) H ( s, ) () d H (, ) [ 0] s (3) H ( s, ) H( s, ) sh ( s, ) (4) 0 s H(, s) e e (5)! Sice the differetiatio operator D satisfies 0 sd H( s, ) e e or euivaletly! De e this ca also be writte as sd H ( s, ) e (6) By (5) we have e s e which by coparig coefficiets is euivalet with the oets s 0 ( )!! (7) The Taylor epasio ( s)!! s ( s) s s s s H(, s) e e e e e D e 0 0 gives the Rodrigues forula s s H (, s) sd e (8) e Let g() deote the ultiplicatio operator with a fuctio g( ), ie g() f ( ) g( ) f ( )

3 - - - Sice Dg( ) g() Dg s s s () we have sde e sd e ad therefore e - s - sd e s sd (9) This iplies that the Herite polyoials H ( s, ) satisfy H (, s) sd (0) A geeraliatio of (0) is Burchall s operatioal forula (cf eg [5]) which iplies Nielse s idetity H-,, () 0 sd ssd H ( s, )!( s) H ( sh, ) ( s, ) () The Rodrigues forula provides a easy proof that for polyoials f ( ) s f f( ) e d s (3) It suffices to show that s H (, s) e d 0 for 0 This is iediate by (8) ad itegratio by parts: s s s H (, s) e d( s) D e d( s) D e 0 For 0 we eed the oraliatio s e s d Fro (4) we see that H (, s) H (, s) s![ ] (4) 3

4 Bivariate -Herite polyoials ad their geeratig fuctios The polyoials I this sectio I freely use soe well-ow otatios of aalysis i the for I used i [7] Let e oly etio that we always assue 0 ad that D deotes the differetiatio operator f ( ) f( ) defied by D f() which satisfies D [ ] with [ ] [ ] ( ) My origial approach to these polyoials has bee (cf [3], [4]) to loo for seueces of oic polyoials which satisfy Dp( ) [ p ] ( ) () ad are orthogoal with respect to a liear fuctioal with c 0 By Favard s theore they ust satisfy a 3-ter recurrece of the for If we apply the operator p ( ) p ( ) c [ ] This eas that p ( ) p ( ) a( ) p ( ) () D ad observe that [ ] p ( ) [ ] p ( ) p ( ) a( ) p ( ) or a ( ) p( ) p ( ) p( ) [ ] Coparig with () we see that a ( ) or [ ] a a a ( ) ( ) [ ] () By choosig a() D f ( ) D f ( ) f ( ) we get s we get the Herite polyoials H ( s,, ) which satisfy H ( s,, ) H( s,, ) sh [ ] ( s,, ) (3) with iitial values H (, s, ) 0 ad H (, s, ) 0 It is easy to deterie the coefficiets of H ( s,, ) Fro (3) it is clear that H s a for soe a (, ) (,, ) (, ) 4

5 Futherore H (0, s, ) s [ ] H(0, s, ) which gives H (0,, ) 0 s ad H 0, s, ( ) s [ ]!! This iplies [ ]! a[, ] [ ]! D H(, s, ) H 0 (0, s, ) ad therefore [ ]! H(, s, ) ( s) []!! It is easily verified that (4) really satisfies both () ad (3) Now we defie the liear fuctioal by (4) H (,, ) [ 0] s By Favard s theore we ow that the seuece H (,, ) s is orthogoal with respect to Sice H ( s,, ) is oic we have H s H s By (3) we see that H( s,, ) s [ ] H ( s,, ) This iplies (,, ) (,, ) H( sh,, ) ( s,, ) s[ ]![ ] (5) Fro (3) we deduce the followig deteriat represetatio s [] s H(, s, ) det [ ] s If we replace by ad observe that we get H ( s,, ) H( s,, ) s [ ] H ( s,, ) (6) ad Sice D D idetity () becoes H(, s, ) ( s) []!! (7) 5

6 DH s,, [ ] H s,, (8) Soe forulae becoe sipler if we cosider the polyoials K(, s, ) H, s, (9) Here we have K, s, ( s) []!! (0) ad K ( s,, ) K( s,, ) sk [ ] ( s,, ) () Furtherore DK ( s,, ) [ K ] ( s,, ) () Rear There are ay orthogoal polyoials which reduce to H ( s, ) for But oly the cotiuous Herite polyoials H ( s,, ) (cf eg [7] ) which satisfy H ( s,, ) H ( s,, ) [ sh ] ( s,, ) see to have iterestig properties too O the other had there are aalogues with siple forulae ad recurrece relatios but which are ot orthogoal For eaple the siplest aalogue of forula () sees to be h (, s, )!!( s) 0 which has bee studied by Kirschehofer [] ad satisfies ad (3) Dh( s,, ) [ h ] ( s,, ), (4) h ( s,, ) h( s,, ) sh [ ] ( s,, ) s [ ]( ) h ( s,, ) (5) h ( s,, ) h( s,, ) sh [ ] ( s,, ) (6) 6

7 Geeratig fuctios I the followig we eed differet aalogues of the epoetial series These are well ow, but for the coveiece of the reader I state the eplicitly The power series ad are related by ad e ( ) (7) [ ]! 0 0 E ( ) (8) [ ]! e ( ) E ( ) (9) e ( ) E ( ) (0) They ca be regarded as foral power series or as coverget power series I the secod case (7) coverges for whereas (8) is a etire fuctio Let us ote that Fro the defiitio of De( a) ae( a), DE( a) ae( a), a a De, ae a a DE ae D these idetities are euivalet with e( ) ( ) e( ), E ( ) ( ) E ( ) () () This iplies the series epasio e (, ), (3) ( ; ) ( ; ) for ad the epasio 7

8 (, ) ( ; ) ( ; ) E (4) which coverges for all I their doais of covergece we have e ( ) e(( ), ) ad E ( ) E(( ), ) Note that e (, ) ;, (5) because ( ; ) ( ; ) e (, ) () E, ; Let us also ote that (cf eg [4]) e ( ) aa a (6) e ( a) [ ]! 0 As a aalogue of (5) we get the geeratig fuctio of the Herite polyoials e ( ) s H(, s, ) e ( ) E (7) 0 [ ]! s e For (4) is euivalet with [ ]! [ ]! [ ]! j j H(, s, ) ( s) []!! 0 j j ad ( s) []!! ( s) [ ]! [] [4] [ ] s ( ) ( ) ( ) ( ) [ ]! s e 8

9 A euivalet versio of the geeratig fuctio for is s ; H s e E s (8) 0 (,( ), ) (, ), ( ; ) ; Note that li H (,( ) s, ) Whereas this is o direct aalogue of the Herite polyoials it is very useful sice the ifiite products o the right-had side are soeties easier to hadle Sice D e ( ) e ( ) we see that (7) is euivalet with sd H(, s, ) E, (9) which geeralies (6) j s Fro (7) we see that e( ) e j(,, ) H s ad therefore j0 [ j]! ( s) [ ]!! H (, s, ) 0 (30) s d e s e we have d s Sice H (, s) ( s) H(, s)!! (3) 0 0 For H ( s,, ) we deduce fro (7) the aalogue s H ( s,, ) H( s,, ) [ ]! ( ) [ ]! (3) 0 0 To prove this let f () f( ) f() Applyig to (8) we get 9

10 H(,( ) s, ) (,( ), ) 0 ; H s 0 ; ; 3 3 s ; s ; s ; s ; ; ; 3 ; s s s H(,( ) s, ) ; ; 0 Replacig ( ) we get (3) s ; I the sae way as above we get K s e E (33) 0 s (,, ) [ ]! ad K (,( ) s, ) es, E, (34) 0 ( ; ) s ; ; I this case K(,( ) s, ) (,( ), ) 0 ; K s 0 ; s ; s ; s 0 ; ; ; s; s; s; s s ; s ( ) K(,( ) s, ) s ; ; Rear For h (, s, ) we get s h( s,, ) e( e ) 0 [ ]! (35) ad thus h s e e s (36) 0 (,( ), ) (, ), ( ; ) ; s ; 0

11 Fro this we see that for the liear fuctioal defied by oets 0 ad s [ ]!! h (,, ) [ 0] s we get the It is easily see that hh 3 [3] s 0, which iplies that the seuece orthogoal h caot be 3 The siplest special cases 3 The polyoials with the siplest right-had side of (8) occur for s Here we get the polyoials h(; ) H,, (37) which have bee called i [3] discrete Herite polyoials I They satisfy h ( ; ) h ( ; ) h ( ; ) (38) with geeratig fuctio for 0, e(, ) ; h ( ; ) (39) ; e ; The polyoials h ( ; ) are give by The first ters are h (; ) () ; (40) 4 ( ; ),, ( ),, ( ) ( ), h These polyoials have first bee cosidered by Al-Sala ad Carlit i [] They studied ore ( a geerally polyoials U ) ( ) with geeratig fuctio 0 U ( a) e(, ) ( ), ; eea (, ) (, )

12 which for a reduce to ( ; ), Fro ; ; ; ee (, )(, ) h because h ( ; ) ; 0 ; ; ; j j ; ; ; ( )( ) ( ) ( ; ) ( ; ) j we deduce by (6) that The idetity (9) becoes j j j ( ; ) j0 j h (4) e, h (, ) (4) where ( D ) with ( ) By (30) we have ; h ( ; ) 0 (43) For the liear fuctioal defied by h ( ; ) [ 0] we get the oets ; 0 (44) 3 Istead of the polyoials h ( ; ) for it is coveiet to cosider the polyoials ( ; ) ;,,,, (,, ) h i h i i H i H K for 0, the so called discrete Herite polyoials II Fro () we see that the polyoials h ( ; ) satisfy h ( ; ) h ( ; ) h ( ; ) (45) (46)

13 By (34) we have 0 These polyoials have the eplicit epressio ( ; ) (; ) ( ; ) ; h (47) h ( ; ) ( ) ; (48) It is easily verified that 3 ; h ( ; ) (49) Therefore the oets with respect to the liear fuctioal L defied by L h ( ; ) [ 0] (50) are L L ; 0 (5) 33 A aalog of (4) for h (,, ) is This iediately follows fro (36) j j h (,, ) ( ) 0 j0 j (3) 3 Rodrigues-type forulae 3 Bivariate ad discrete - Herite polyoials I 3 For the bivariate Herite polyoials we get two differet aalogues of (0) The first oe is H (, s, ) sd sd sd (3) This follows fro () ad (3) which iply H ( s,, ) H (, s, ) sd H (, s, ) sd H (, s, ) 3

14 The secod oe is 3 H (, s, ) sd sd sd (33) To prove this we observe that H ( s,, ) H( s,, ) sh [ ] ( s,, ) ad DH ( s,, ) [ ] H( s,, ) iply H ( s,, ) H ( s,, ) ( ) [ ] H( s,, ) H( s,, ) ad therefore H s H s sh s H s (,, ) (,, ) [ ] (,, ) (,, ) H s sh s (,, ) [ ] (,, ) Chagig we get aother recurrece relatio for the Herite polyoials H ( s,, ) H, s, [ sh ], s, (34) Sice H( s,, ) H, s, this is euivalet with H ( s,, ) H s,, sh [ ] s,, (35) This iplies H( s,, ) H ( s,, ) sdh ( s,, ) sd H ( s,, ) 3 sd sd sd (34) also iplies that H (, s, ) sd (36) Let us give aother forula for the right-had side of this idetity The differetiatio operator D satisfies D f ( ) g( ) f ( ) D g( ) D f ( ) g( ) (37) ad thus D D f() = f() + D f () (38) 4

15 ad D f() = f() D + D f () (39) a a Sice DE ae we get fro (38) E -a -a + + ( sd) E sd as (30) ad therefore for a s H (, s, ) ( sd ) E ( s ) E ( s ) (3) Lettig i (3) s( ) s ad a ( s ) we get E ( sd ) E H(, ( ), ) s s s or euivaletly ( ) s D ; H(,( ) s, ) s s ; (3) 3 By (3) we have (; ) ( ) ; ; h D (33) Let us give aother proof 4 Sice( D ) ; ; 5

16 we get by (37) ( ) D ; ; ; ( ) D or D ( ) D ; ; (34) This iplies (33) Observig that D D this forula iplies [3], (389) D ; h( ; ) ; h ( ; ) (35) 3 The discrete -Herite polyoials II By (36) for we deduce the followig aalogue of (0): K (, s, ) sd (36) a For f( ) e [] we have Df() f() D a or a a e De D a [] [] (37) Coparig (37) with (36) we get the Rodrigues-type forula - - e e []s K (, s, ) sd ( s) D []s (38) Sice sd we get i the sae way ; ; s s 6

17 - - ; ( sd) sd s - - ; s ad therefore the euivalet forula K(, s, ) sd ( s) ( ) s ; D (39) ( ) s ; For s this reduces to the Rodrigues type forula [3], (390) (; ) ( ) h D ; ; (30) Rear For the polyoials h (, s, ) we get (cf []) by (4) ad (6) h (, s, ) s D (3) The seuece of polyoials h (,, ) s is the ubral-iverse seuece to the seuece (,, ) satifies (,, ) H s This eas that the liear ap U defied by U h s s D U H s or with other words that (,, ) H s D (3) The proof is straightforward by iductio sice (3) hold for 0 ad : H s D, s, s D [ ] s I aother for this fact has already bee show i (30) The Herite polyoials satisfy (cf eg [5]) H ( sh, ) ( y, s ) ( y ) (33) 0 A aalogue is y H(, s, ) h( y, s, ) (34) 0 0 7

18 This follows fro e ( ) s e( e ) ( y) e( ) (,, ) (,, ) ye H s h ys s 0 [ ]! 0 [ ]! e by coparig coefficiets 33 A -Burchall forula I [6] I have give soe aalogues of Burchall s forula The siplest oe is the followig Burchall forula (,, ) (35) 0 3 sd sd sd H s sd The proof is by iductio 0 0 s H s sd D s D H s sd sd H s 0 ( ) ( )( ) H (, s, ) sd H(, s, ) sd sd H(, s, ) sd H(, s, ) sd ( ) (,, ) (,, ) 0 0 ( ) (,, ) sd H(, s, ) sd H (, s, ) sd H s sd ( ) ( ) (,, ) If we apply (35) to,, H s we get a aalogue of Nielse s idetity H ( s,, ) [ ]! ( s) H ( sh,, ), s, (36) ( ) H, s, H, s, Sice this coicides with H( s,, ) [ ]!( s) H( sh,, ), s,, (37) which has bee proved i [4] with aother ethod 8

19 4 Associated probability easures 4 The -itegral I the classical case the liear fuctioal which satisfies s f f( ) e d s H (, ) [ 0] s is give by I order to fid a aalogue we eed the Jacso itegral (cf [0], [], [4]) We assue that 0 We call F( ) a atiderivative of f ( ) if DF( ) f( ) This eas that F( ) F( ) ( ) f ( ) or ( ) F ( ) ( ) f ( ) or F( ) ( ) f ( ) ( ) f ( ) ( ) f ( ) 0 0 If this su coverges absolutely it is clear that F( ) is a atiderivative of f ( ) because F( ) F( ) ( ) f ( ) I the classical case all atiderivatives of 0 are costats I the case also each fuctio ( ) with ( ) ( ) is a atiderivative of 0 But it ca be show that up to a costat ay fuctio has at ost oe atiderivative that is cotiuous at 0 (Cf [] for details) This leads to the followig defiitio We always assue that 0 Let 0 a b The defiite itegral is defied as ad b 0 j0 j j j f ( d ) f b b (4) b ( ) b ( ) a ( ) a 0 0 f d f d f d (4) provided that the sus coverge absolutely, for eaple if f ( ) for soe 0 j j j j For the we have f M M ( ) b Note that f ( d ) depeds o the values of f ( ) i the whole iterval (0, b ] a If f ( ) is cotiuous at 0 ad the itegral coverges absolutely, the b M i a eighbourhood of 0 D f ( ) d f ( b ) f ( a ) (43) a For all atiderivatives of D f( ) which are cotiuous at 0 are of the for f ( ) C j 9

20 Fro the product rule ( ) ( ) ( ) ( ) ( ) ( ) D f g f D g g D f we obtai by (43) the forula for itegratio by parts b b f ( ) Dg ( ) d f( bgb ) ( ) f( aga ) ( ) g ( ) Df ( ) d (44) a a 4 The probability easure for the discrete -Herite polyoials I We start fro the Rodrigues-type forula h(; ) ( ) D ; ; By ultiplyig both sides with ; ad applyig we see that D ; is of the for Ch ; ; ad thus vaishes at This iplies that for 0 h ( ; ) ; d 0 (45) Sice ; is cotiuous ad therefore bouded o [,] the itegral (45) is give by a absolutely coverget series Thus the liear fuctioal which gives h ( ; ) [ 0] is f f ( ) ; d ( ) ; d (46) Let us ow calculate ; d ; d 0 By defiitio of the itegral we have ; d ( ) ; ( ) ; 0 ( ) ; e, ( ) ; ( ; ) ; 0 ;

21 As Euler has show 4 6 ( ) ; ( ) ( ) Therefore d 0 If we choose ; ( ) ; ( ; ) ; i Jacobi s triple product idetity ( ) ( ; ) ; ( ; ) we see that Thus we get ; d ( ) ; ; ( ) (47) 0 f( ) ; d j j j j ; d j0 f( ) f( ) f( ) ; 0 (48) Let us loo what this gives for the oets Observe that j0 For odd it is clear that 0 ( ) j () j j ; ; ; ; ; ; e ; ; ; ; ; ; ; 0 ; This iplies () j j ; ; j0 0 which agrees with (44)

22 Rear Forula (48) has already bee foud by Al-Sala ad Carlit [] They observed that the oets satisfy ; ( ) ad 0 ad therefore are characteried by with iitial values ad 0 ad costructed a ifiite su with the sae properties This is siply the reverse of the coputatio above 43 Probability easures for the discrete -Herite polyoials II I this case we eed the otio of a iproper itegral We defie for c 0 N c c jn j j N ( ) ( ) li ( ) ( ) li 0 0 N N j 0 j j j j li ( ) fcc ( ) fcc N jn j f d f d f c c provided the last su coverges absolutely The forula b b f ( ) Dg ( ) d fbgb ( ) ( ) f( aga ) ( ) g ( ) Df ( ) d (49) a a also holds i the iproper case by lettig N b c tedig to ifiity I a siilar way we defie c c j j j f ( d ) ( ) ( ) f c f c c (40) j By (30) we see that (; ) ( ) h D ; ; (4) This iplies that for 0

23 c h ( ; ) ( ) ( ) 0 c c d D D c ( ; ; ; c c if the itegral eists ad the liits are 0 j j For this it suffices to show that cf( c) coverges if j f ( ), ; j j It is clear that cf( c) coverges because f( ) Sice j0 ; we have c cf c c c N ( ) N N N N N N j j ad therefore we see that also cf ( c) coverges j0 Therefore i this case the liear fuctioal L has ifiitely ay represetatios as a itegral I order to copute j c c ( ) j d ( ) ; c c c j j ; c ; c ; j j we eed Raauja s suatio forula (cf [], 053) a; ; ; a ( a ; ) ; ; a This gives c c ( ) c ; ; ; ( ) c d c ( ; c ; j ; j j c ; ; ; c Cobiig these results we see that the liear fuctioal L defied by represeted by each of the iproper itegrals (cf [3]) L h ( ; ) [ 0] ca be 3

24 c ; ; ; c c f( ) L f( ) d c (4) ( ) c ; ; ; ; c As above we ca copute L ;, (43) which ca also be obtaied fro (49) 5 Soe curious idetities The taget ubers T ad Euler ubers E ca be defied by e e e e ( )! ad ( ) T 0 ( ) E e e ( )! 0 The first ters are T ad E,,6, 7, 7936, 0,,5,6,385, 0 The polyoials H (, s ) are polyoials i s with degree o the polyoials i s defied by Cosider the liear fuctioal F F H (, s) [ 0] (5) The F H (, s) ( ) T (5) ad E ( )!! F s (53) 4

25 To prove this let 0 s H() H(,) s e! The H ( ) e H( ) ad therefore Applyig F we get e e H (, s) H (, s) ( )! e e ( )! 0 0 e e (, ) ( ), ( )! ( )! F H s T 0 e e 0 which gives (5) To prove (53) observe that s s F e F e e e FH(, s) e FH (, s) 0! 0 ()! e e e ( ) T e ( ) E 0 ()! e e e e 0 ( )! These arguets ca iediately be trasferred to the polyoials H (, s, ) Defie the taget ad Euler ubers by e( ) e( ) ( ) T ( ) e ( ) e ( ) []! ad 0 ( ) E( ) e ( ) e ( ) [ ]! 0 Let ow be the liear fuctioal defied by The ad H (, s, ) [ 0] (54) H T ( ) (55) s E ( ) []!! (56) 5

26 Refereces [] Waleed A Al-Sala ad Leoard Carlit, Soe orthogoal -polyoials, Math Nachrichte 30 (965), 47-6 [] George E Adrews, Richard Asey ad Raja Roy, Special fuctios, Cabridge Uiv Press 999 [3] Joha Cigler, Operatorethode für Idetitäte, Moatsh Math 88 (979), [4] Joha Cigler, Eleetare -Idetitäte, Sé Loth Cob B05a (98) [5] Joha Cigler, Proble 493, Aer Math Mothly 7(3), 00 [6] Joha Cigler, Soe operator idetities related to Herite polyoials, arxiv:00630 [7] Joha Cigler, Cotiuous -Herite polyoials: A eleetary approach, arxiv: [8] Jacues Désaréie, Les -aalogues des polyoes d Herite, Sé Loth Cob B06b (98) [9] Thoas Erst, Sur les polyoes -Herite de Cigler, Algebras, Groups ad Geoetries 7(00), -4 [0] George Gasper ad Mia Raha, Basic hypergeoetric series, Cabridge Uiv Press 990 [] Victor Kac ad Poa Cheug, Quatu Calculus, Spriger 00 [] Peter Kirschehofer, Bioialfolge, Shefferfolge ud Fatorfolge i der Aalysis, Situgsber ÖAW, ath-at Kl II, 88 (979), [3] Roelof Koeoe ad Reé F Swarttouw, The Asey-schee of hypergeoetric orthogoal polyoials ad its -aalogue, [4]To H Koorwider, Special fuctios ad coutig variables, arxiv:-alg/ [5] Gia-Carlo Rota, Fiite operator calculus, Acadeic Press 975 6