Some q-analogues of Fibonacci, Lucas and Chebyshev polynomials with nice moments

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1 Some -aaloges o Fiboacci, Lcas ad Chebyshev polyomials with ice momets Joha Cigler Faltät ür Mathemati, Uiversität Wie Ui Wie Rossa, Osar-Morgester-Platz, 090 Wie ohacigler@ivieacat Abstract We give a srvey o some iterestig aaloges o Fiboacci, Lcas, Chebyshev ad related polyomials ad their momets 0 Itrodctio The momets o the Fiboacci polyomials or eivaletly o the Chebyshev polyomials o the secod id are (mltiples o) the Catala mbers C ad the momets o the Lcas polyomials or eivaletly o the Chebyshev polyomials o the irst id are (mltiples o) the cetral biomial coeiciets B I wat to show how these acts geeralize to varios aaloges It trs ot that the ow aaloges o Fiboacci ad Lcas polyomials are ot eivalet with the ow aaloges o Chebyshev polyomials ad have ite dieret properties The aaloges o the Chebyshev polyomials are orthogoal polyomials ad their momets are mltiplies o c ( ) [ ] ad ( ) b respectively The same act holds or a crios class o o-orthogoal Fiboacci ad Lcas polyomials, whereas the momets C ( ) o the Carlitz Fiboacci polyomials - which are orthogoal - do ot have explicit expressios Istead o that the geeratig ctio C ( ) C ( ) has a simple E( ) represetatio i the orm C ( ) or some aaloge o the expoetial series E( ) It is iterestig that or some extesios o the Chebyshev polyomials similar expressios or the geeratig ctios exist I the irst sectio I recall some well-ow acts abot Fiboacci ad Lcas polyomials ad some coseeces o their orthogoality The I cosider the o-orthogoal aaloges where other methods are eeded I the third sectio we cosider the Chebyshev polyomials where most reslts o the classical case have simple coterparts Fially we stdy the momet geeratig series or some extesios o the Chebyshev polymials 0

2 Bacgrod material Let the special Fiboacci polyomials ( x ) be deied by ( ) ( ) 0 x x () They satisy the recrrece relatio ( x) x ( x) ( x) () with iitial vales ( x) ad ( x) x 0 ad are orthogoal with respect to the liear ctioal deied by ( ) x [ 0] More precisely we have The momets x This gives x 0 ad ( x) ( x) [ m] (3) m ca easily be dedced rom the ormla x ( x) 0 (4) x C, (5) where C is a Catala mber The momet geeratig ctio is 4 x C( ) (6) 0 ad satisies C ( ) C ( ) (7) Eivaletly we have C ( ) x x 4 x x 4 I we set ad, the x ad x Note that x ad Sice satisies recrsio () ad the iitial vales we get the Biet ormlae ( x) (8) (9)

3 Let s also cosider a variat l ( x ) o the Lcas polyomials deied by 0 0 (0) l( x) ( ) x l ( x) The polyomials l ( x ) satisy the recrrece relatio with 0 ad or 0 l ( x) xl ( x) l ( x) () We have l( x) ( x) ( x) or ad l( x) ( x) or 0, This implies or 0 the Biet ormlae l ( x) () The polyomials l( x ) are orthogoal with respect to the liear ctioal l deied by l ( x) l [ 0] More precisely ad l l x or 0 ( ) 0 From the represetatio l ( x) l ( x) [ m] (3) l m we dedce that the momets l x are l x x l( x) 0 (4) 0 ad where l x, is a cetral biomial coeiciet (5) The momet geeratig ctio is ( ) (6) 4 l x B 0 Eivaletly B ( ) (7) 3

4 3 Let s ow state some well-ow otatios ad reslts or idetities which will be eeded later (c eg [5]) We always assme that 0 x; x 0 Let x; xx x ; ; ; Let [ ] [ ] ad The biomial coeiciets satisy ad ad or 0 Let be the operator ( x) ( x) ( x) ( x) Let D be the dieretiatio operator deied by D ( x) x x The ( ) xd ( ( ) xd ) x x x x x sice A simple aaloge o the biomial theorem is the act that the so called Rogers-Szegö polyomials r ( x, s) x s 0 ca be represeted as r ( x, s) ( x s ) This ollows by idctio becase ( xs ) xs x s xs xs xs Note also that Dr ( xs, ) [ r ] ( xs, ) becase Dr ( xs, ) Dxs [ x ] s [ ] x s [ r ] ( xs, ) Thereore the Rogers-Szegö polyomials satisy the recrsio r ( x, s) ( xs( ) sxd ) r ( x, s) ( xs) r ( x, s) ( ) xsr ( x, s) (8) We shall also eed the ollowig hypergeometric versio o the biomial theorem (c eg [3]) a ; ax; x (9) ; x ; or x 0 4

5 Notes The polyomials ( x ) are the special case ( x) Fib ( x, ) o the bivariate Fiboacci 0 which satisy polyomials Fib ( x, s) s x Fib ( x, s) xfib ( x, s) sfib ( x, s) with iitial vales Fib ( x, s) 0 ad Fib ( x, s) 0 For 0 the polyomials l ( x ) are the special case l ( x) Lc ( x, ) o the bivariate Lcas polyomials which satisy Lc ( x, s) 0 or 0 0 Lc ( x, s) s x ad There is also a close coectio with the Chebyshev polyomials The Chebyshev polyomials o the irst id T ( x ) satisy T( x) xt ( x) T( x) with iitial vales T ( x) ad T( x) x ad Lc ( x, ) 0 T( x) Lcx, 4 The moic Chebyshev polyomials o the irst id 0 4 (0) t( x) x or 0 ad t 0 ( x) satisy t( x) xt ( x) t( x) with 0 ad 4 The Chebyshev polyomials o the secod id U( x ) satisy U( x) xu ( x) U( x) with iitial vales U ( x) ad U ( x) x ad 0 U ( x) Fib ( x, ) Fib x, 4 The moic Chebyshev polyomials o the secod id satisy ( x) x ( x) ( x) 4 0 () 4 ( x) x Idetity (4) is a special case o the ollowig sitatio: Let p ( x ) be polyomials satisyig p ( x) xp ( x) p ( x) () ad iitial vales p ( x) ad p ( x) x 0 These are moic orthogoal polyomials 5

6 The there are iely determied coeiciets c (, ) sch that x c (, ) p( x) (3) This implies c (, ) p( x) x c (, ) p( x) c (, ) p ( x) p ( x) p( x) c(, ) c(, ) Thereore we get 0 c (, ) c (, ) c (, ) (4) with iitial vales c(0, ) [ 0] ad bodary vales c (, ) 0 I we apply the liear ctioal we get x c (5) (,0) Let s recall a well-ow combiatorial iterpretatio o (4): The mber cm (, ) is the weight o all lattice paths i which iitial poit 0,0 ad edpoit m,, where each step is either a p-step,, or a dow-step,, The weight o a p-step is ad the weight o a dow-step with edpoit, is The weight o a path is the prodct o the weights o its steps The weight o a set o paths is the sm o their weights The trivial path 0,0 0,0 is also cosidered ad has by deiitio weight It is the clear that c,0 0 Each otrivial path has a ie decompositio o the ollowig orm: a p-step, a maximal path which ever alls der height ad eds o height, a dow-step to height 0 ad a rther path which eds at height 0 Thereore we get 0 0 c,0 b(,0) c,0, (6) where bm (, ) is the correspodig weight whe is replaced by Let ( ) c(,0) ad ( ) b(,0) The (6) is eivalet with ( ) 0( ) ( ) (7) For the polyomials ( x ) we have ad thereore (7) redces to (7) Idetity (4) is eivalet with c(, ) ad (, ) 0 c The correspodig matrix c (, ) is ow as Catala triagle (c [3], OEIS, A053) 6

7 For the polyomials l( x ) we have 0 ad Thereore ( ) C( ) ad ( ) ( ) C( ), which gives ( ) C( ) 4 Some crios -aaloges Let s cosider the polyomials ( x, ) ( ) x 0 () They are ot oly aaloges o ( x ) i the sese that lim ( x, ) ( x) bt ca be obtaied rom ( x ) by irst comptig the operator ( x ( ) D) ad the applyig it to the costat polyomial Ths ( x, ) ( x( ) D ) () They satisy the recrrece relatio ( x, ) x( D ) ( x, ) ( x, ) (3) with iitial vales ( x, ) ad ( x, ) x 0 They also satisy ( x, ) x ( x, ) x ( x, ) ( x, ) (4) 3 4 Let, ( x, ) [ 0] The polyomials (, ) x are ot orthogoal For example 3 (, ) ( ) 0, x3 x Nevertheless there a ice aaloge o (4): This implies that the momets, x x ( x, ) 0 (5) are where, x c ( ), c ( ) [ ] is a explicit aaloge o the Catala mbers (6) 7

8 I do ot ow a simple aaloge o the geeratig ctio (7), bt we have istead which is a aaloge o (8) 0 ( ) c (7) ; Let ow or ad l 0 ( x, ) ad l ( x, ) x The or 0 ad l 0 ( x, ) The polyomials l ( x, ) satisy l ( x, ) ( x, ) ( x, ) (8) [ ] l( x, ) ( ) x 0 [ ] (9) l ( x, ) x( ) D l ( x, ) t l ( x, ) (0) with iitial vales l 0 ( x, ) ad l ( x, ) x This ca also be writte as l ( x, ) l ( x( ) D ) () Also the polyomials l ( x, ) are ot orthogoal The idetity x l( x, ) 0 () implies that l, x, i we deie the liear ctioal l, by l, ( x, l ) 0 I do ot ow a simple aaloge o (6) or b ( ) 0 Istead o this is a aaloge o 0 ( ) (3) 0 ; (4) 0 8

9 Proos ad remars The polyomials ( x, ) ad l( x, ) have bee systematically stdied i [8] To prove () it sices to compare coeiciets i (3) Sice these polyomials are ot orthogoal ad ths do ot satisy a 3 term recrrece o the orm () the above combiatorial iterpretatio ails Bt ormla (3) implies Biet-type ormlae or these polyomials: Let A be the operator A x( ) D For each polyomial px ( ) i x we deie px pa ( ) Ths ( x, ) ( x) l ( x, ) l ( x) ad aalogosly or 0 This is a exact versio o a symbolic method which I sed i [9] This implies x 0 (5) From (8) we get x 0 Sice these are by idctio polyomials i x we get agai by idctio Ax x x 0 To prove () observe that or odd l( x, ) I m the x m m m m l x x m m m m m m m m m (, ) Sice l( x, ) ( x, ) ( x, ) we also get (5) 9

10 Comparig coeiciets we see that (7) is eivalet with 0 ( ) c ( ) [ 0] Bt this is is clear sice ( ) c ( ) ( ) x ( x, ) [ 0],, 0 0 I the same way idetity (4) is eivalet with 0 ( ) ( x, ) 3 Special -Chebyshev polyomials 3 The polyomials x (, ) ( ) 0 x (3) ; ; will be called special Chebyshev polyomials o the secod id They satisy the recrrece relatio ( x, ) x ( x, ) ( x, ) (3) with iitial vales ( x, ) ad ( x, ) x 0 The polyomials ( x, ) are orthogoal with respect to the liear ctioal deied by, ( ( x, )) [ 0] (33) More precisely we have ( ( x, ) ( x, )) [ m] ; ;, m (34) From the momets are give by ( x, ) x 0 ; ; (35) 0

11 , x ( ) ( ) ( ) [ ] C (36) ; ; where C ( ) is a Catala mber i the sese o Adrews [] As aaloge o C C C(, ) C ( ) 0 we get or the geeratig ctio Let C(, ) C(, ) C(, ) C (, ) (37) ( ) ; h ( ): This is a aaloge o sice ; The is a aaloge o 0 C 4 ; hh ( ) ( ) ; h ( ) C (, ) ( ) (38) 3 The special Chebyshev polyomials o the irst id are the polyomials [ ] 0 [ ] ; ; (39) t ( x, ) ( ) x The polyomials t ( x, ) satisy the recrrece relatio with ( ) 0 ad ( ) t ( x, ) xt ( x, ) ( ) t ( x, ) (30) The polyomials t( x, ) are orthogoal with respect to the liear ctioal t, deied by More precisely we have or 0 The idetity t, m t, t( x, ) [ 0] (3) t ( x, ) t ( x, ) [ m] ; ; (3) x 0 ; ; t( x, ) (33)

12 implies the momets ; t, x (34) Let By the biomial theorem (, ): ( ) ; 0 ; G (35) G (, ) g ( ) G (, ) 0 ; (36) Note that g ( ) is a aaloge o sice G (, ) ; implies gg ( ) ( ) The geeratig ctio o the momets is t, x g 0 (, ) G ( ) (37) G (, ) Notes I [0] we itrodced bivariate Chebyshev polyomials T ( x, s, ) o the irst id T( xs,, ) xt ( xs,, ) st ( xs,, ) by with iitial vales T ( x, s, ) ad T( x, s, ) 0 x ad bivariate Chebyshev polyomials U ( x, s, ) o the secod id U( xs,, ) xu ( xs,, ) su ( xs,, ) by with iitial vales U ( x, s, ) ad U ( x, s, ) ( ) x 0 We the have or 0 ad T ( x,, ) t ( x, ) (38) ; U( x,, ) ( x, ) ; Similar polyomials have also appeared i other pblicatios, c [0] or [] ad the literatre cited there They are related to the Al-Salam ad Ismail polyomials itrodced i [] (39)

13 The recrrece relatios ca be easily veriied by comparig coeiciets Proos ca also be od i [0] or [] The same holds or (35) ad (33) Formla (38) ollows rom the biomial theorem (9) sice ; ; ; h ( ) ( ) ; ; ; 0 0 ( ) [ ] C ; ; 0 0 C(, ) C(, ) h h ( ) (38) implies ( ) ( ) C(, ) C (, ) ad ths (37) The biomial theorem gives ; 0 0 ; g ( ) ( ) ; ; ; Note that the biomial theorem gives G (, ) ; ; 0 Ths by compariso o coeiciets (36) is eivalet with the well-ow ormla 0 (30) 4 Momet geeratig ctios s 4 Cosider the orthogoal polyomials () with (,, zs) ( z)( z) Callig them ( xzs,,, ) we get ( s) ( x, z, s, ) x 0 z z (4) ; ; s Note that ( x, z, s,) Fib x, ad that ( x,,, ) ( x, ) ( z) 3

14 These polyomials are also related to the Al-Salam ad Ismail polyomials (c [0] or [] ad the literatre cited there) By (7) we get or the geeratig ctios ( zs,,, ) c(,0, zs,, ) ad ( zs,,, ) b(,0, zs,, ) s ( zs,,, ) ( zs,,, ) ( zs,,, ) ( z)( z) s Sice ( z, s, ) (,, ) (,, ) z s z s ( z)( z) (, z, s, ) (, z, s, ) (, z, s, ) we have Thereore ( zs,,, ) satisies For this gives s (, z, s, ) (, z, s, ) (, z, s, ) ( z)( z) s ( zs,,,) (,,,) zs ( z) s (,,,) ( z) zs C ad ths (4) (43) Uortately i the geeral case there are o simple ormlae or c(,0, z, s, ) Bt there is a simple represetatio or their geeratig ctios Cosider the series ad Fz (,, ) (44) z ; ; 0 Gz (,, ) (45) z ; ; 0 which are aaloges o the expoetial series 0 The we ca write the geeratig ctio i the orm! z 4

15 s F, z, z G( s, z, ) ( zs,,, ) s G( s, z, ) F, z, z (46) For the special case z, s we get aother represetatio o the geeratig ctio o the Adrews Catala mbers:,, (,, ) F G C (, ),,, (47) F (,, ) G(,, ) This is eivalet with the ollowig two (dieret) expressios or C ( ) ; 0 ( ) C ( ), ( ) C ( ) ( ) ; 0 (48) For z 0 ad s we get the Carlitz Fiboacci polyomials (c eg [7]) 0 (49) ( x,0,, ) ( ) x I this case Their geeratig ctio C ( ) C ( ) satisies Comparig coeiciets this gives 0 0 C ( ) C ( ) C ( ) (40) C ( ) C ( ) C ( ) with C ( ) 0 Frther properties ca be od i [] I [6] I proved with aother method ( ) C( ) 0 0 (4) which gives a recrrece to compte C ( ) 5

16 This idetity ollows immediately rom orthogoality becase 0 F, ( ) C ( ) x F ( x, ) 0 The same reasoig gives i the geeral case c (,0, zs,, ) ( s) 0 0 z z (4) ; ; As a special case o (46) we get the well-ow reslt E( ) C ( ), E( ) (43) where 0 E( ) G(,0, ) is a aaloge o the expoetial series ; 4 I we choose we get the polyomials Note that t ( x, ) l ( x,, ) 0 (, z) z (, z) z z 0 z ; z ; (44) z l ( x, z, ) ( ) x (45) For the geeratig ctio we get l( z,, ) l( z,, ) l( z,, ) ad z l(, z, ) l(, z, ) l(, z, ) ( z)( z) 6

17 This implies ad G l( z,, ) ( z,,, ) G G, z, l ( z,, ) G, z, For the special case t ( x, ) l ( x,, ) we get same as (37) 3, z,, z, G,, l (,, ) G,, (46) (47) which is the Proos ad Remars Let F( z,, ) ( z,, ) (48) F( z,, ) The z ( z,, ) ( z,, ) ( z,, ) ( z)( z) (49) This ollows rom Fz (,, ) Fz (,, ) ( z,, ) zz Fz (,, ) Fz (,, ) ( z ; ) ( ; ) F, z, 0 z z ( z,, ) ( z,, ) ( z)( z) F, z, ( z)( z) Sice the ormal power series ( z,, ) is iely determied by (49) we see that s F, z, s z zs,,,, z, z s F, z, z Next observe that lim, z,, G(, z, ) series i, becase or 0 as a ormal power 0 z ; ; lim ; It is easily veriied that ( xzs,,, ) also satisies the recrsio s ( xzs,,, ) x ( xzs,,, ) ( xzs,,, ) (40) ( z)( z) Thereore 7

18 (, z, s, ) s (, z, s, ) (, z, s, ) (, zs,, ) ( z)( z) (, zs,, ) (, zs,, ) For this gives (,, ) (,, ) (,, ) G s z s G s z G s z G( s, z, ) ( z)( z) G( s, z, ) G( s, z, ) ad ths G( z, z, ) ( z,, ) G( z, z, ) Gz (,, ) We ow give aother proo Let ( z,, ) Gz (,, ) The (, z, ) (, z, ) (, z, ) ( z)( z) (4) Sice ( ) ( z z Gz (,, ) Gz (,, ) z; ; z; ; z; ; ( z) z ( z) z we get z ; ; G, z,,, ( z,, ) G, z, ( z)( z) G, z, G z G z,, G, z, G, z, ( z,, ) ( z,, ) ( z)( z) G, z, G, z, ( z)( z) I the same way we see that z ( z,, ) ( z,, ) ( z,, ) ( z)( z) (4) is eivalet with G( z, z, ) ( z,, ) (43) G( z, z, ) Comparig (49) ad (4) we that F(, z, ) G( z, z, ) (44) F( z,, ) Gzz (,, ) 8

19 The ctioal eatio ( z,, ) ( z,, ) ( z,, ) z G, z, is eivalet with ( z,, ) G, z, To prove this observe that,,,, Gz Gz ( ) z ; ; z z; ; z G z,, This implies 3 3 G, z, G, z, G, z, G, z, ( z,, ) G, z, z G, z, z G, z, z z z (,, ) (,, ) G, z, Remar I their paper [4] M J Catero ad A Iserles prove that the ratioal ctios a ( z, ) deied by a ( z, ) ad 0 a (, z ) (45) ; z ; ; z ; or 0 satisy 0 z lim a( z, ) ( ) C ( z) This reslt also ollows rom or cosideratios (46) For the eatios (45) are eivalet with ad ths with 0 F(, z, ) a ( z, ) F( z,, ) a (, z ) ( ) 0 0 z)(; ) 0(; z)(; ) Now we have F (, z, ) (,, ) F z F( z,, ) z Fz (,, ) becase 9 (47)

20 Fz (,, ) Fz (,, ) Fz (,, ) z z; ; z Ths we get 0 0 z ; ; z ; ; z ; ; F(, z, ) a ( z, ) (, z, ) z F(, z, ) z By (49) z z ( z,,) ( z,,) C ( z) ( z) which implies (46) Reereces [] WA Al-Salam ad MEH Ismail, Orthogoal polyomials associated with the Rogers- Ramaa cotied ractio, Paciic J Math 04 (983), [] G E Adrews, Catala mbers, -Catala mbers ad hypergeometric series, J Comb Th A 44 (987), [3] G E Adrews, R Asey ad R Roy, Special ctios, Ecyclopedia o Math ad Appl 7, 999 [4] M J Catero ad A Iserles, O a crios -hypergeometric idetity, Noliear Aalysis, Spriger Optimizatio ad Its Applicatios 68 (0) -6 [5] J Cigler, Elemetare -Idetitäte, Sém Loth Comb B05a (98) [6] J Cigler, Operatormethode ür -Idetitäte V: -Catalabäme, Sitzgsber ÖAW 05 (996), 75-8 [7] J Cigler, -Fiboacci polyomials, Fib Qart 4 (003), 3-40 [8] J Cigler, A ew class o -Fiboacci polyomials, Electroic J Comb 0 (003), R 9 [9] J Cigler, -Lcas polyomials ad ad associated Rogers-Ramaa type idetities, arxiv: [0] J Cigler, A simple approach to -Chebyshev polyomials, arxiv:04703 [] J Cigler, -Chebyshev polyomials arxiv: [] J Fürliger ad J Hobaer, -Catala mbers, J Comb Th A 40 (985), [3] OEIS, The Olie-Ecyclopedia o Iteger Seeces, 0