Hankel determinants of some polynomial sequences. Johann Cigler

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1 Hael determiats of some polyomial sequeces Joha Cigler Faultät für Mathemati, Uiversität Wie Abstract We give simple ew proofs of some Catala Hael determiat evaluatios by Ömer Eğecioğlu ad Alesadar Cvetović, Predrag Raović ad Miloš Ivović ad prove aalogous results for sums of momets of symmetric orthogoal polyomials Itroductio Let a ( ) be a give sequece ad let rx (, ) ( ) 0 ax or 0 rx (, ) xa ( ) a ( ) Cosider the Hael matrices A a( i ), 0, i (, ) i, 0 R r i x ad let dx (, ) det R Suppose that d ( ) det A 0 for all dx (, ) We wat some iformatio about the ratio d ( ) Ömer Eğecioğlu [0] has show that if we choose for a ( ) the Catala umbers C or the cetral biomial coefficies B ad let dx (, ) dx (, ) rx (, ) a ( x ), the the quotiet of the determiats is give by 0 d ( ) d (,0) dx (, ) i i i ( ) x d (,0) i0 i Alesadar Cvetović, Predrag Raović ad Miloš Ivović [9] have show that det Ci Ci is a Fiboacci umber We show more geerally that aalogous results hold if the umbers a ( ) are momets of symmetric orthogoal polyomials p( x, ) By symmetric we mea that they satisfy a recurrece of the form p( x) xp ( x) tp( x) Some of these results have also bee obtaied with other methods i [3], [] ad []

2 The polyomials rx (, ) a ( x ) 0 dx (, ) d (,0) We start with the obvious fact that RA A short ispectio shows that RA i det (, ) i i h i with h (,0) (0,0), 0 i xh ad h(, i ) xh(0, ) for i ad i i h (, i ) x xh (0, ) if i For let A u i (, ) The i, 0 ai ( u ) (, ) [ i ] ad 0 h (, i ) r( i, x) u(, ) 0 Note that h (0, ) r(, x) u(, ) Therefore 0 i i i h ( i,0) ri ( xu, ) (,0) u (,0) xrx (, ) a ( i ) x xh (0,0) If i we get i the same way i i i h ( i, ) r( i, x) u(, ) u(, ) x r(, x) a( i ) x x h (0, ) For i the secod sum for i adds i x For example for 3 the matrix is (we write h( ) i place of h (0, ) ) 3 RA h(0) h() h() h(3) xh(0) xh() xh() xh(3) xh(0) x xh() xh() xh(3) xh(0) x xh() x xh() xh(3) 3 3 The special form of the matrix implies that det RA h(0,0) () Cosider polyomials p ( x ) which satisfy a recurrece of the form p ( x) xp ( x) t p ( x) () for some umbers t 0 with iitial values p ( x) 0 ad p ( ) 0 x These polyomials have the form

3 (3) 0 p ( x) ( ) v(, ) x ad are orthogoal with respect to some liear fuctioal, ie pp 0 m for m We call them symmetric orthogoal polyomials This liear fuctioal is uiquely determied by ( p ) [ 0] If the sequece t t is give we say that the polyomials p ( ) x are associated with t Throughout this paper we assume that a ( ) is of the form a ( ) ( x ) (4) for a sequece of symmetric orthogoal polyomials We shortly call a ( ) symmetric momets For more iformatio o orthogoal polyomials we refer to [4] Cotiuig the recurrece p ( x) xp( x) t p( x) xp( x) t xp( x) t3p3( x) we see that p ( x) p( x) t p( x) t t3p4( x) ( ) t t3 tp0( x) x This implies that p ( x) ( ) t t3 t x By orthogoality x p ( ) x for Therefore M x ( ) p ( x) ( ) v x satisfies ( ) ( ) (, ) t t3t x t t3t 0 (5) for 0 m m x M x m ( ) [ 0] (6) This meas that ( ) v(, ) a( m) [ m 0] t t t 0 3 (7) 3

4 A Thus the traspose of the first colum of is u(0,0), u(,0),, u(,0) ( ) v(, ) t t t 3 ad () implies that ( ) det RA h(0,0) ( ) v(, ) r(, x) t t3t 0 We state these results i the followig Lemma Let p( x) ( ) v(, ) x satisfy the recurrece p( x) xp ( x) tp( x) with 0 p ( x) 0 ad p ( x) ad let 0 a ( ) ( x ) ad rx (, ) a ( x ) The dx (, ) ( ) ( ) v(, ) r(, x) (8) d (,0) t t3t 0 If A( ) is defied by A( ) a( ) ad A() 0, ad if R( x, ) A ( x ), the R( x, ) rx (, ) ad therefore H(0,0) u(,0) R(, x) ad H (0,0) u(,0) R(, x) coicide with h (0,0) u(,0) r(, x ) 0 Therefore the correspodig quotiets of Hael determiats satisfy D x D x d x D(,0) D(,0) d(,0) (, ) (, ) (, ) (9) We ca ow prove Theorem Let t t with t 0 0 for all ad let T T t Let 0 p (, ) 0 x t be the orthogoal polyomials associated with the sequece t ad p( x, T ) the orthogoal polyomials associated with T The the quotiets of the Hael determiats of the momets correspodig to t are give by dx (, ) ( ) d (,0) ttt 3 p ( xt, ) (0) Proof Sice p( xt, ) xp ( xt, ) tp( xt, ) we get v (, ) v (, ) t v (, ) () 4

5 We assert that ad ( ) v(, ) r(, x ) p( x, T) 0 () ( ) v(, ) r(, x ) xp ( x, T) 0 (3) This obviously is true for 0 Therefore by iductio ( ) v(, ) r(, x ) ( ) v(, ) r(, x ) t ( ) v(, ) r(, x ) xp ( x, T) 0 t ( ) v(, r ) (, x) xp ( xt, ) t p ( xt, ) p ( xt, ) ad ( ) v(, ) r(, x ) 0 v r x t v r x 0 0 ( ) (, ) (, ) ( ) (, ) (, ) ( ) v(, ) a( ) x ( ) v(, ) r(, x ) 0 0 ( ) (, ) (, ) (, ) 0 t v r x x p x T t xp ( xt, ) xp ( xt, ) Here we used the fact that ( ) v(, ) a( ) 0 by (7) 0 Remars The polyomials P( x, t) p( x, t) are orthogoal with respect to the liear fuctioal L defied by LP [ 0] They satisfy the recurrece ad satisfy L x a( ) with P( xt, ) xs P ( xt, ) U P ( xt, ) (4) S t 0 0 S t t for 0 U t t (5) 5

6 Theorem gives oly the ratio of determiats but ot d (,0) But i our cotext d (,0) is implicitly ow because d (,0) UU 0 U (cf eg [5]) Now we cosider some iterestig Examples ) Defie the bivariate Fiboacci polyomials F ( x, s ) by F( x, s) xf ( x, s) sf( x, s) with iitial values F ( x, s) 0 ad F( x, s) 0 ad cosider the polyomials I this case t for all Therefore T t ( ) (, ) ( ) 0 p x F x x (6) It is well ow that the momets are the Catala umbers x C (7) Thus we get Corollary (Ömer Eğecioğlu [0]) For a ( ) C the Hael determiats are dx d (,0) (, ) ( ) F ( x, ) ( ) x ( ) x 0 0 (8) ) Choose t 0 ad t for 0 The ( ) (, ) ( ) for 0 0 p x L x x (9) Here L ( x, s ) are bivariate Lucas polyomials defied by L( x, s) xl ( x, s) sl( x, s) with iitial values L ( x, s) ad L( x, s) x Note that 0 p0( x) L0( x, ) I this case x (0) 6

7 ad T Therefore we get Corollary (Ömer Eğecioğlu [0]) For a ( ) the quotiets of the Hael determiats are also give by dx d (,0) (, ) ( ) F ( x, ) ( ) x ( ) x 0 0 () 3) Defie the bivariate (Carlitz -) q Fiboacci polyomials F ( x, s, q ) by F( xsq,, ) xf ( xsq,, ) q sf ( xsq,, ) () 3 with iitial values F ( x, s, q) 0 ad F( x, s, q) 0 They satisfy F ( x, s, q) s q x 0 Let ow t q The 0 (3) p( x) F ( x,, q) ( ) q x I this case the momets x C( q) (4) are the q Catala umbers C( q ) of Carlitz whose geeratig fuctio f ( z) C ( ) q z 0 satisfies f ( z) zf( z) f( qz) (See eg [4]) 3 Sice t t3 t q q ad t q we have p( xt, ) F ( x, qq, ) This implies Corollary 3 For a ( ) C( q) the quotiets of the Hael determiats are dx d (, ) ( ) F ( x, q, q) ( ) q x (,0) 0 (5) q Remar Christia Krattethaler [] (upublished) has previously proved Corollary 3 with aother method 7

8 4) Defie the ( qb, ) Fiboacci polyomials F ( x, b, s, q ) by the recursio 3 q s F ( x, b, s, q) xf ( x, b, s, q) F ( x, b, s, q) q bq b with iitial values F ( x, b, s, q) 0 ad F( x, b, s, q) 0 These are variats of the Al Salam ad Ismail polyomials ([]) (6) The (cf eg [7]) x s q 0 F ( x, b, s, q) q Let t ( q )( q ) qb qb The correspodig orthogoal polyomials are x 0 p ( x, t) F ( x,,, q) ( ) q q q (7) Sice t q ( q )( q ) 3 we get p( xt, ) F ( x, q, qq, ) The momets are the q Catala umbers of George Adrews ([]) q [ ] q q x (8) A proof ca be foud i [6] This implies Corollary 4 For the (Adrews-) q Catala umbers we get q a ( ) [ ] q q 8

9 dx (, ) ( ) (,,, ) q F x q q q d (,0) q q ( ) q x 0 (9) 5) Cosider the geeralized q Lucas polyomials (cf [7],[8]) They satisfy [ ] 0 [ ] L ( x, s, q) q s x q q q q q s L ( x, s, q) xl ( x, s, q) L ( x, s, q) with iitial values L ( x, s, q) ad L( x, s, q) x 0 The correspodig orthogoal polyomials p( x ) are defied by p ( x) ad 0 p( x) L( x,, q) for 0 The momets are (cf [6]) x q The correspodig t are t 0 q ad t I this case tt 3t q q Therefore p( xt, ) F ( x,, qq, ) q ( ) q q ad t q ( q ) q (30) (3) (3) This implies Corollary 5 For a ( ) q the quotiets of the Hael determiats are dx (, ) ( ) d (,0) q (,,, ) q F x q q (33) 9

10 6) For t q a ad t q b the orthogoal polyomials are (cf [4] ) ad b p( x, t) ( a) q x 0 0 a (34) b (, ) ( ) 0 0 a (35) p x t a q x Therefore the Hael determiats for the momet sequece satisfy dx d (,0) (, ) (36) ( ) q x q a b 0 0 For q some of the momets are well-ow For ( ab, ) (,) the momets are the little Schröder umbers,, 3,, 45,97, ad for ( ab, ) (,) we get the (large) Schröder umbers,,6,,90,394, (Cf [6] ad OEIS [3] A00638 ad A00003) Corollary 6 Let a ( ) be the sequece of little Schröder umbers ad t (,,,,,, ) The dx d (, ) ( ) (, ) ( ) p x T x (,0) 0 0 If a ( ) is the sequece of large Schröder umbers the (37) dx d (,0) (, ) ( ) p( x, t) ( ) x 0 0 (38) Remars It is clear that for each sequece T there are may sequeces t such that the divided Hael dx (, ) ( ) determiats are p( xt, ) d (,0) tt 3t It suffices to choose t 0 0 arbitrary 0

11 It should be oted that ot every sequece with o-vaishig Hael determiats ca be represeted as momets of symmetric orthogoal polyomials For example the sequece,,,4,9,,5,7, of Motzi umbers M (cf OEIS [3], A00006) satisfies det M i, 0 for all The first divided Hael determiats dx (, ) for the Motzi i d (,0) 4 6 umbers tur out to be, x, x, x x x, d(, x ) Thus is a polyomial of degree istead of degree 4 Thus M caot be of the d(,0) form x correspodig to orthogoal polyomials of the form () So what ca be said about the ratios of Hael determiats of the Motzi umbers? To aswer this questio we eed the followig fact (cf eg [4]) Let f (, zu) M () uz satisfy u f (, z u) uzf(, z u) z f(, z u) ( ) ad defie the liear fuctioal u F x M u The the correspodig orthogoal polyomials satisfy F by P( x, t) ( xu) P ( x, t) P ( x, t) (39) 0 Thus P( x, t) F ( xu, ) (40) For u we get the Motzi umbers M () M I this case there are o t satisfyig F ( u, ) (5) But for geeral u it is easily see that t t F ( u, ) t ad Therefore (0) implies dx d (,0) (, ) ( ) F ( u, ) p ( x, T) (4) Now observe that the polyomials p (, ) xt satisfy both p ( xt, ) xp( xt, ) tp ( xt, ) ad p ( xt, ) x u p ( xt, ) p ( xt, ) 3 This implies that F ( u, ) p x T F x u F x u (, ), (, ) F ( u, ) Therefore (4) gives dx (, ) ( ) F( u, ) F ( xu, ) F( u, ) F( xu, ) (4) d (,0)

12 By cotiuity this relatio also holds for u Observig that the sequece F (, ) 0,,,0,,, is periodic with period 6 we see that i this case 0 ad d(3, x) F ( x, ) F ( x, ) (43) 3 3 d(3, x) d(3, x) F ( x, ) (44) 3 The case rx (, ) ax ( ) a ( ) Geeralizig the results of Alesadar Cvetović, Predrag Raović ad Miloš Ivović [9] I have show i [5] a geeral result which i the preset termiology ca be stated i the followig way: Theorem Let The rx (, ) ax ( ) a ( ) (45) det ri ( x, ) i, 0 det ai ( ) i, 0 p ( x, t) (46) ad det ri (, x) i, 0 p3( x, t) det ai ( ) i, 0 x (47) Proof Here I give a simpler proof With the otatio itroduced i (3) we get m m 0 x p ( x) ( ) v(, ) x Applyig the liear fuctioal to this idetity we get ( ) ( ) (, ) ( ) ( ) (, ) ( ) a m v a m v am for 0 m This implies that 0 ri ( x, ) bi (, ) ai ( ) i, 0 i, 0 i, 0

13 with x x 0 bi (, ) i, ( ) v(, ) ( ) v(, ) ( ) v(, ) x v(,) Therefore i, 0 0 det bi (, ) ( ) v(, ) x p ( x) Idetity (47) follows aalogously from m p3( x) 4m x ( ) v(3, ) x, x 0 which for 0 m implies a ( m) ( ) v(3, am ) ( ) 0 Note that (46) for 0 p(0, t) ( ) t0t t 0 det ai ( ) 0 sice x implies that i, 0 Examples ) For a ( ) C we have p( x, t) F ( x, ) This implies the results obtaied by Alesadar Cvetović, Predrag Raović ad Miloš Ivović [9] Observe that F( x, ) F( x,) Therefore (46) reduces to det ri (, x) i, 0 F 3( x, ) ( ) F 3( x,) det ai ( ) i, 0 This implies for 0 ad for x that x that Ci det where F deotes a Fiboacci umber I the same way (47) gives i, 0 det C C F (,) F, (48) i i i,

14 i i i, 0 4 det C C F (49) ) As aother simple example cosider the Hermite polyomials H ( x ) defied by H ( x) xh ( x) ( ) H ( x) The momets are x Therefore we get ( )!! (cf eg [4]) ad det (i )!!( x i ) i, 0 det (i )!! i, 0 H ( x) (50) det (i )!!( x i 3) i, 0 H3( x) det (i )!! i, 0 x (5) 3) More iterestig are the umbers ( ) det M i( u) Mi( u) have bee computed i [3] These results are also simple cosequeces of Theorem M u The determiats det M i ( u) Mi ( u) ad By (40) we ow that p( x, t) P( x, t) F ( xu, ) Therefore det xm i ( u) M i ( u) i, 0 F ( x u, ) (5) det M ( u) For x 0 this gives For x we get from (5) Note that for ad F (, ) i i, 0 det M ( u) ( ) F ( u, ) F ( u, ) (53) i i, 0 i i i, 0 det M ( u ) M ( u ) F ( u, ) (54) u the sequece F (, ) 0,,,0,,, is periodic with period 6 0 For the secod formula observe that p ( xt, ) xp( xt, ) t p( xt, ) ad therefore F ( u, ) p ( x, t) F ( u, ) p ( x, t) F ( u, ) p ( x, t) which gives by iteratio 3 4

15 3 0 0 F ( u, ) p ( x, t) ( ) F ( u, ) p ( x, t) ( ) F ( u, ) F ( x u, ) Therefore we get det xm ( ) ( ) i u M i u i, 0 ( ) ( ) (, ) (, ) det F u F xu (55) M (, ) 0 ( ) F u i u i, 0 For u this does ot mae sese for all sice F3 (, ) 0 But sice for geeral u det M ( u) F ( u, ) we get by multiplyig (55) with F (, ) u i i, 0 i i i, 0 0 det xm ( u) M ( u) ( ) F ( u, ) F ( x u, ) (56) Sice both sides are polyomials i u this also holds for u 4) Fially I state the correspodig results for the little Schröder umbers s ( ) which follow from (34) ad (35) These results have also bee obtaied i [] with aother method i, 0 x 0 0 i, 0 det xs( i ) s( i ) ( ) det si ( ) (57) The first terms are x x x x x x 3, 4, 7, i, 0 x 0 0 i, 0 det xs( i ) s( i ) ( ) det si ( ) (58) The first terms are x x x x x x 3 3, 6 7, 9 3 5, Refereces [] Waleed A Al-Salam ad Mourad EH Ismail, Orthogoal polyomials associated with the Rogers-Ramaua cotiued fractio, Pacific J Math 04 (983), [] George E Adrews, Catala umbers, q-catala umbers ad hypergeometric series, J Comb Th A 44 (987), [3] Naiomi T Camero ad Adrew CM Yip, Hael determiats of sums of cosecutive Motzi umbers, Li Alg Appl 434 (0), 7-7 5

16 [4] Joha Cigler, Eiige q-aaloga der Catala-Zahle, Sitzugsber ÖAW 09 (000), 9-46, [5] Joha Cigler, Some relatios betwee geeralized Fiboacci ad Catala umbers, Sitzugsber ÖAW (00), [6] Joha Cigler, Hael determiats of Schröder-lie umbers, arxiv: [7] Joha Cigler, A simple approach to q Chebyshev polyomials, arxiv: [8] Joha Cigler, q-chebyshev polyomials, arxiv: [9] Alesadar Cvetović, Predrag Raović ad Miloš Ivović, Catala Numbers, the Hael Trasform, ad Fiboacci Numbers, Joural of Iteger Sequeces 5 (00), Article 03 [0] Ömer Egecioglu, A Catala-Hael determiat evaluatio, Cogressus Numeratium 95 (009), [] Se-Peg Eu, Tsai-Lie Wog ad Pei-La Ye, Hael determiats of sums of cosecutive weighted Schröder umbers, Li Alg Appl 437 (0), [] Christia Krattethaler, persoal commuicatio [3] OEIS (The O-Lie Ecyclopedia of Iteger Sequeces), [4] Gérard Vieot, Ue théorie combiatoire des polyômes orthogoaux gééraux, Motréal 983 6