Abstract. 1. Introduction This note is a supplement to part I ([4]). Let. F x (1.1) x n (1.2) Then the moments L x are the Catalan numbers

Transcription

1 Abstract Some elemetary observatios o Narayaa polyomials ad related topics II: -Narayaa polyomials Joha Cigler Faultät für Mathemati Uiversität Wie ohacigler@uivieacat We show that Catala umbers cetral biomial coefficiets ad Narayaa polyomials are momets of aalogues of Fiboacci ad Lucas polyomials ad related polyomials Itroductio This ote is a supplemet to part I ([4]) Let ( ) ( ) x F x () be Fiboacci polyomials ad defie a liear fuctioal L by The the momets L x are the Catala umbers L x L F ( ) [ ] x () C This well-ow fact is the special case t of the followig result (cf [4] ad the literature cited there): The Narayaa polyomials (3) C() t t (4) ca be represeted as momets Lx of the liear fuctioal L defied by L F ( ) [ ] x t where F ( x t) ( ) t x (5) are geeralized Fiboacci polyomials which satisfy the recurrece

2 F ( x t) xf ( x t) F ( x t) F ( x t) xf ( x t) tf ( x t) (6) Oe of our purposes is to give a ice aalogue of this result More precisely we defie ice polyomials F ( x t ) such that the liear fuctioal defied by LF ( ) x t [ ] has as momets the Narayaa polyomials Lx C( t ) t [ ] (7) which for t reduce to the Catala umbers C( ) C( ) [ ] (8) We will always suppose that ad use the otatios [ ] [ ] ad ; ; ; for It is well ow that for each seuece a with a such that all Hael determiats det ai there are uiuely defied moic polyomials p ( ) i x of degree which are orthogoal with respect to the liear fuctioal L defied by L x a Orthogoality meas that L ppm for m ad L p Sice p ( ) implies that Lp [ ] Moreover there are uiuely determied umbers a ( ) such that x this x a ( ) p( x) (9) By applyig L we see that a ( ) L x a For the polyomials F ( x t ) are the orthogoal polyomials with momets ( ) ad L x C x t L x Let ow L be the liear fuctioal defied by L x C ( t ) ad L x It would be temptig to cosider also i this case the correspodig orthogoal polyomials but it seems that there is o simple explicit formula or recurrece for them ot eve for t

3 I that case the first orthogoal polyomials are x x x 3 x x 4 4 x x x x 4 4 The first umbers a ( ) are Fortuately there also exist ice polyomials with the same momets Let us cosider first the case The orthogoal polyomials for the liear fuctioal L defied by L x C They satisfy L x are the Fiboacci polyomials F ( x) ( ) x a( ) F ( x) with a( ) ad x ad a ( ) else Let me setch how to fid a ice aalogue of this situatio It is easier to begi with a ( ) A atural aalogue is [ ] a( ) [ ] first terms are ad a ( ) else The Now we are looig for the polyomials F ( x ) such that x a( ) F ( x ) () 3

4 Their coefficiets are give by the iverse matrix ai ( ) Fortuately this also turs out to be ice The first terms are Thus the seuece F ( x ) x 3 x x begis with 4 x x x 5 3 x x x F ( ) x ca of course also be computed iductively by It is ow easy to guess that i geeral F ( x ) x a( ) F ( x ) Remar ( ) ( ) F x x () Note that these Fiboacci polyomials which have bee cosidered i [] ad [3] are ot orthogoal There are also ice orthogoal aalogues of the Fiboacci polyomials ( ) x Their momets are the ie the Carlitz Fiboacci polyomials Carlitz Catala umbers but ufortuately these have o closed formula I this ote we first recall some results about the above metioed class of o- orthogoal Fiboacci ad Lucas polyomials whose momets are Catala umbers ad cetral biomial coefficiets ad the propose ice aalogues of the geeralized Fiboacci ad Lucas polyomials of part I such that the correspodig momets are Narayaa polyomials C( t ) t [ ] ad M( t ) t 4

5 Some bacgroud material Let us first state some ow results (cf [][][3]) As already metioed the Fiboacci polyomials satisfy F ( x ) ( ) x () x F ( x ) () If we defie a liear fuctioal L by LF x ( ) [ ] the we get Lx C( ) (3) [ ] where C( ) is a aalogue of the Catala umbers [ ] C The Lucas polyomials for ad L ( x ) satisfy [ ] L ( x ) ( ) x [ ] (4) x L ( x ) (5) If we defie the liear fuctioal M by M L ( ) [ ] x the we get M x M( ) is a cetral biomial coefficiet (6) It is easy to verify that L x F x F x (7) ( ) ( ) ( ) for Moreover L( x ) F( x ) ad L( x ) F( x ) x 5

6 These results ca be proved with a iversio formula by L Carlitz [] Aother proof is i [] ad [3] Carlitz uses the fact that for mi( ) c ( ): ( ) (8) To prove this let U be the liear operator o ad U x defied by x x U for itegers ( ): ( ) The c U U U U because U Lemma (Carlitz [] Theorem 7) If u ( ) v ( ) the ( ) ( ) ( ) v u Proof ( ) u( ) ( ) v( ) v ( ) ( ) v ( ) v ( ) ( ) ( ) v ( ) by (8) If we choose u ( ) x we get () By (7) ad 6

7 we get L( x ) F( x ) F( x ) ( ) ( ) F x F x x ad thus also (5) Remar The simplest recursio for F ( x ) for a fixed umber x is (cf []) F ( x ) xf ( x ) xf ( x ) F ( x ) (9) Let us also ote that comparig coefficiets gives the recursio F ( x ) xf ( x ) F x () We will also cosider the polyomials ad P( x ) F x ( ) x () F x ( ) ( ) Q x x x () For these polyomials we get ad x P ( x ) (3) x Q ( x ) (4) 7

8 If we defie liear fuctioals L ad L by the (3) ad (4) give Aalogously let L P ( x ) [ ] L Q ( x ) [ ] L x C ( ) (5) (6) Lx C ( ) ad [ ] [ ] R( x ) L x ( ) x (7) L x [ ] S( x ) ( ) x x [ ] (8) This implies Let M R x ad ( ) [ ] x R ( x ) x S ( x ) M S ( x ) [ ] The we get (9) By comparig coefficiets we get M M x x () R x Q x Q x () ( ) ( ) ( ) 8

9 3 -Narayaa polyomials as momets I the followig we exted the above results by itroducig a ew parameter t as i part I I [4] we have defied F ( x t ) by the recursios F ( x t) xf ( x t) F ( x t) F ( x t) xf ( x t) tf ( x t) ad iitial values F ( x t) ad F( x t) x (3) If L deotes the liear fuctioal defied by LF x t ( ) [ ] the we have L x C () t ad L x (3) where C() t t (33) for ad C () t is a Narayaa polyomial Theorem Let () t ad () t t ad defie F ( x t ) by the recursio F ( xt ) xf ( xt ) ( tf ) ( xt ) (34) with iitial values F ( x t ) ad F( x t ) x These polyomials are explicitly give by ( ) ( ) ( ) ( ) ( ) ( ) F x t t x F x t t x If L deotes the liear fuctioal defied by LF x t where for is a Narayaa polyomial ( ) [ ] the we have (35) L x C ( t ) ad L x (36) C t t [ ] (37) Proof To prove (35) we have oly to show that (34) holds This follows by comparig coefficiets 9

10 a t we get by usig For a a ad the Vadermode formula ( ) ( ) ( ) ( )( ) ( ) ad aalogously ( ) This implies that ( ) F ( x ) F ( x ) (38) To prove (36) let at ( ) be the uiuely defied umbers such that By (34) we get atf ( ) ( xt ) x (39) a( t ) a( t ) ( t) a( t ) (3) with iitial values a ( t ) ad a( t ) [ ] This implies that ( ) a( t ) t ( ) a( t ) t ad at ( ) else (3) I order to show this we must verify that ad a( t ) a( t ) a( t ) (3) a( t ) a( t ) ta( t ) (33)

11 (3) follows from a( t ) a( t ) t t t a( t ) (33) follows from ( ) a( t ) ta( t ) t ( ) t t t a( t ) As special case we get for a( t ) t C ( t ) (34) C ( ) t is related to the Catala umbers c ( ; ) of J Fürliger ad J Hofbauer [5] They have show that C( ) C( ) [ ] This result follows agai from Theorem because of (38) ad (3) (35) Remar Let us also cosider the polyomials P( xt ) F xt F x t Q x t x ad

12 Corollary Let ad The with Q ( x t ) ( ) t x (36) B ( t) t (37) B ( tq ) ( xt ) x (38) t B( t) t C [ ] (39) Note that by [5] C C( ) i accordace with (6) B ( t ) ca also be writte as [ ] B ( t) t [ ] For t B ( t ) reduces to B Corollary Let ( ) ad The with ( ) (3) P ( x t ) ( ) t x A t t ( ) (3) A ( t ) P( x t ) x (3)

13 A( t ) t C ( t ) [ ] (33) The first terms of A ( t ) are -Narayaa polyomials of type B I [4] we have see that the orthogoal polyomials L ( x t ) whose momets are the Narayaa polyomials M() t t of type B satisfy the recurrece with iitial values L ( x t) ad L( x t) x Here we have L ( x t) xl ( x t) ( t) L ( x t) (34) () t t t () t t t t () t t for We ow show that there exists a atural aalogue of L ( x t ) with L( x t) L( x t) ad which for t reduces to the Lucas polyomials: Theorem Let ( t) t t ( t) t t t ( t) t (35) ad defie L ( x t ) by 3

14 with L ( x t ) ad L( x t ) x L( xt ) xl ( xt ) ( tl ) ( xt ) (36) Let M deote the liear fuctioal defied by M L ( ) [ ] x t the we get M x M ( t ) t (37) Proof Let at ( ) satisfy at ( ) a ( t ) ( ta ) ( t ) (38) with a ( t ) ad a( t ) [ ] The (36) implies that By iductio it is easy to verify that atl ( ) ( xt) x (39) a t t ( ) ( ) ( ) ( )( ) ( ) t ( ) ( ) t t t a t t t ad at ( ) else If M L ( ) [ ] x t the (39) implies M x a( t ) t M( t ) (33) By Vadermode we see that a( ) ad a( ) Comparig with (9) we coclude that L ( x ) is a Lucas polyomial Further it is clear that L ( x t) L ( x t) Let R xt L xt ( ) I order to get a formula for R ( xt ) observe that 4

15 B ( t ) D ( t ) td ( t ) (33) which is euivalet with the easily verified idetity [ ] [ ] By (38) this implies D ( t ) Q ( x t ) tq ( x t ) D ( tq ) ( xt ) td ( tq ) ( xt ) B ( tq ) ( xt ) x Comparig with (337) we see that if we let Q ( x t ) Q ( x t ) The first terms of ( ) R xt Q xt tq xt (33) R x t are ( ) ( ) ( ) From (33) ad (36) we get the formula with ( ) ( ) ( ) (333) R x t c t x ( ) ct ( ) t for (334) ct ( ) t This ca also be writte as R x t t x t ( ) ( ) ( ) ( ) (335) 5

16 By (39) ad (33) we get Corollary Let The ( ) D ( t ) a( t ) t (336) Let M be the liear fuctioal defied by D ( t ) R( x t ) x (337) M R ( x t ) [ ] The M x D ( t ) B ( t ) t (338) L xt Let ow S( x t ) x From (36) we get for Corollary Let t t S( x t ) t R x t R x x t (339) The E t a t t t (34) Let ( ) ( )( ) ( ) ( ) t M be the liear fuctioal defied by E ( t ) S( x t ) x (34) M S( x t ) [ ] The t M M x E( t ) t t t (34) 6

17 Remar For the umbers D ( t ) there exists a aalogue of the Catala-Stieltes matrix for orthogoal polyomials: ad D ( t) ( td ) ( t ) td ( t ) (343) D ( t ) D ( t ) t D ( t ) td ( t ) (344) Let us metio some curious coectures: Let t f() t operator with respect to the variable t The f () t f( t) ( t ) be the differetial m m m m m m t D ( x t) R( x t ) [ m]! cm t m x (345) Here c ( t m ) for c ( ) t m ad for For m we get m [ m] c ( t m ) t m m [ m] (346) c( t ) t C t [ ] Thus t D ( x t) R( x t ) C t x (347) It should be oted that the umbers are the coefficiets of the powers mm m c ( t m) t m m Cxt ( ) m of the geeratig fuctio Cxt ( ) C( tx ) of the Narayaa polyomials (cf [4]) Other such idetities are t B ( tq ) ( xt ) C t x t A ( tp ) ( xt ) C t x (348) (349) 7

18 m m m m m m m t A ( xt ) P ( x t ) [ ] c m tm x (35) m m m m m m t B xt Q x t c m tm x ( ) ( ) [ ] (35) Refereces [] L Carlitz Some iversio formulas Red Circ Palermo (963) [] J Cigler A ew class of Fiboacci polyomials Electr J Comb (3) R 9 [3] J Cigler Lucas polyomials ad associated Rogers-Ramaua type idetities arxiv: 9765 [4] J Cigler Some elemetary observatios o Narayaa polyomials ad related topics arxiv:655 [5] J Fürliger ad J Hofbauer Catala umbers J Comb Th A 4 (985)