q-chebyshev polynomials

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1 -Chebyshev polyomials Joha Cigler Faultät für Mathemati, Uiversität Wie Abstract I this overview paper a direct approach to Chebyshev polyomials ad their elemetary properties is give. Special emphasis is placed o aalogies with the classical case. There are also some coectios with taget ad Geocchi umbers. 0. Itroductio Waleed A. Al Salam ad Mourad E.H. Ismail [] foud a class of polyomials which ca be iterpreted as aalogues of the bivariate Chebyshev polyomials of the secod id. These are essetially the polyomials U( x, s, ) which will be itroduced i (.). I [] I also cosidered correspodig Chebyshev polyomials T ( x, s, ) of the first id which will be defied i (.6). Together these polyomials satisfy may aalogues of well-ow idetities for the classical Chebyshev polyomials T( x) T( x,,) ad U( x) U( x,,). For some of them it is essetial that our polyomials deped o two idepedet parameters. This is especially true for (.36) which geeralizes the defiig propertyx x T( x) U ( x) x of the classical Chebyshev polyomials. Aother approach to uivariate aalogues of Chebyshev polyomials has bee proposed by Natig Ataishiyev et al. i [], (5.3) ad (5.4). I our termiology they cosidered the moic versios of the polyomials T x,, ad U x,,. Sice x U( x, s, ) s U,, s ad x T( x, s, ) s T,, their defiitio also leads to the s same bivariate polyomials (,, ) T x s ad U ( x, s, ). Without recogizig them as aalogues of Chebyshev polyomials some of these polyomials also appeared i the course of computig Hael determiats as i [7] ad [3]. The purpose of this paper is to give a direct approach to these polyomials ad their simplest properties. 00 Mathematics Subect Classificatio: Primary 05A30; Secodary B83, 33C47 Key words ad phrases: Chebyshev polyomials, taget umbers, Geocchi umbers, orthogoal polyomials, Hael determiats.

2 . Some well-ow facts about the classical Chebyshev polyomials Let me first state some well-ow facts about those aspects of the classical Chebyshev polyomials (cf. e.g. [5]) ad their bivariate versios for which we will give aalogues. The (classical) Chebyshev polyomials of the first id T ( x ) satisfy the recurrece T ( x) xt ( x) T ( x) (.) with iitial values T ( x) ad T( x) x. 0 For x this reduces to T (). (.) The (classical) Chebyshev polyomials of the secod id U ( x ) satisfy the same recurrece U ( x) xu ( x) U ( x) (.3) but with iitial values U ( x) 0 ad U ( x), which gives U ( x) x. 0 As special values we ote that U (). (.4) These polyomials are related by the idetity x x T ( x) U ( x) x, (.5) which i tur implies T ( x) ( x ) U ( x). (.6) Remar. For x cos idetity (.5) becomes cos isi cos isi T cos iu (cos )si or euivaletly T (cos ) cos si( ) U(cos ). si (.7) This is the usual approach to the Chebyshev polyomials. Idetity (.6) reduces to cos si. (.8)

3 Ufortuately it seems that this aspect of the Chebyshev polyomials has o simple aalogue. The Chebyshev polyomials are orthogoal polyomials. As is well-ow (cf. e.g. [4]) a p( x) of polyomials with p ( x) ad deg 0 p is called orthogoal with seuece 0 respect to a liear fuctioal o the vector space of polyomials if p p 0 for m. The liear fuctioal is uiuely determied by p [ 0]. Here [ P ] deotes the Iverso symbol defied by [ P] if property P is true ad [ P] 0 otherwise. The values ( x ) are called momets of. Let P ( x ) deote the moic polyomials correspodig to p( x ) ad a (, ) be the uiuely determied coefficiets i ap (, ) ( x) x. (.9) ( x P ( x)) The a (,0) ( x) ad more geerally a (, ). ( P( x) ) 0 By Favard s theorem there exist umbers s ( ), t ( ) such that the three-term recurrece m P( x) ( xs( )) P ( x) t( ) P ( x) (.0) holds. Therefore the coefficiets a (, ) satisfy a(0, ) [ 0] a (,0) s(0) a (,0) t(0) a (,) a (, ) a (, ) s( a ) (, ) t( a ) (, ). (.) This ca be used to compute the momets a (,0) ( x). If the momets are ow the the correspodig orthogoal polyomials P ( x ) are give by 0 x x x x x x x P x x x x x x x x x 3 ( ) det. i det( ( x )) i, 0 (.) So the owledge of the polyomials P ( x ) is euivalet with the owledge of s ( ) ad t ( ) ad this is agai euivalet with the owledge of the momets. 3

4 T ( x) Sice dx cos( ) d [ 0] x for the polyomials T ( x ) the correspodig 0 liear fuctioal L is give by the itegral px ( ) Lp( x) dx (.3) x ad LT The correspodig momets are if 0 T ( x) dx x if 0 (.4) Lx ( ) (.5) x x ad Lx ( ) 0. For the polyomials U ( x ) we get from U ( x) x dx si(( ) )si d [ 0] 0 that the correspodig liear fuctioal M satisfies M ( p) p( x) x dx (.6) ad M U ( ). U x x dx (.7) The correspodig momets are M( x ) x x dx (.8) ad M x ( ) 0. As already metioed i the itroductio for our aalogues we eed bivariate Chebyshev polyomials. 4

5 The bivariate Chebyshev polyomials T ( x, s ) of the first id satisfy the recurrece T ( x, s) xt ( x, s) st ( x, s) (.9) with iitial values T ( x, s) ad T( x, s) x. 0 Of course T ( x) T ( x, ). They have the determiat represetatio x s x s x 0 0 T ( x, s) det x s x (.0) The bivariate Chebyshev polyomials of the secod id U ( x, s ) satisfy the same recurrece U ( x, s) xu ( x, s) su ( x, s) (.) but with iitial values U ( x, s) ad U ( x, s) x. 0 Their determiat represetatio is x s x s x 0 0 U( x, s) det x s x (.) These polyomials are coected via This also implies x x s T ( x, s) U ( x, s) x s. (.3) T ( x, s) ( x s) U ( x, s) ( s). (.4) The Chebyshev polyomials are itimately related with Fiboacci ad Lucas polyomials 0 (.5) F x, s s x 5

6 ad L xs, F ( xs, ) sf ( xs, ) sx (cf. e.g. [0]). Here as usual L ( x, s). 0 (.6) 0 More precisely the moic polyomials T ( x, s) ad T ( x, s) 0 for 0 coicide with the modified Lucas polyomials * L ( x, s) * s L x,. (.7) 4 They are defied by L * ( x, s) L( x, s) for 0 ad (, ) 0 s s recurrece with s ( ) 0, t(0) ad t ( ) for 0. 4 * L x s ad satisfy a three-term The momets ca be obtaied from the formula * ( s) L ( x, s) x. 0 (.8) The moic polyomials U( x, s) are Fiboacci polyomials U( x, s) F ( x, s) s F x,. 4 (.9) s I this case the correspodig umbers s ( ) ad t ( ) are s ( ) 0 ad t ( ). 4 Here the momets ca be obtaied from ( s) F ( x, s) x. 0 (.30) We shall also give aalogues of the followig idetities which express Chebyshev polyomials of odd order i terms of Chebyshev polyomials of eve order: T ( x) ( ) t x T( x) 0 (.3) 6

7 ad U ( x) ( ) G( x) U( x). 0 (.3) Here the taget umbers t,,6, 7,7936, 0 G ad the Geocchi umbers 0,,,3,7,55, 073, are give by their geeratig fuctios 0 ad z z e e t ( ) z z z (.33) e e 0 ( )! z z e e G z ( ) z. z z (.34) e e ( )! 0 Note that t G. (.35). -aalogues We assume that is a real umber. All idetities i this paper reduce to ow idetities whe teds to. We assume that the reader is familiar with the most elemetary otios of aalysis (cf. e.g. [5]). The biomial coefficiets [][] [ ] with [ ] satisfy the recurreces [] [ ] [] [ ]. If we wat to stress the depedece o we write [ ] ad respectively. (.) We also eed the Pochhammer symbol biomial theorem i the form or euivaletly ad the ( ; ) ( )( ) ( x x x x) x x 0 ( ; ) ( ) (.) (, ) ( )( ) ( ). 0 p x y x y x y x y x y (.3) 7

8 We deote by z ez ( ) the expoetial fuctio. It satisfies [ ]! 0 0 z. e( z) [ ]! Sice the Chebyshev polyomials are special cases of Fiboacci ad Lucas polyomials it would be temptig to loo for aalogues related to the simplest aalogues of Fiboacci ad Lucas polyomials (cf. e.g. [0]) 0, F ( x, s, ) s x [ ] L( xs,, ) F ( xs,, ) sf ( xs,, ) sx, 0 [ ] Fib ( x, s, ) s x 0 ad [ ] Luc( xs,, ) Fib ( xs,, ) sfib ( xs,, ) sx. 0 [ ] But here we have o success. Though the polyomials F (,, ) x s are orthogoal there are o closed forms for their momets. Noe of the other classes of polyomials satisfies a 3-term recurrece. So they caot be orthogoal. But it is iterestig that for Fib (,, ) x s ad Luc( x, s, ) the followig aalogues of (.8) ad (.30) ad * ( s) Luc( x, s, ) x 0 (.4) ( s) Fib ( x, s, ) x 0 (.5) hold (cf. [8], (3.) ad (3.)). Notice that Luc ( x, s, ) ad Luc * ( x, s, ), whereas * 0 0 Luc( x, s, ) Luc( x, s, ) for 0. Fortuately there do exist aalogues of the recurreces (.9) ad (.) which possess may of the looed for properties. 8

9 Defiitio. The Chebyshev polyomials of the first id are defied by the recurrece T( xs,, ) ( ) xt ( xs,, ) st ( xs,, ) (.6) with iitial values T ( x, s, ) ad T ( x, s, ) x. 0 3 The first terms are, x,[] x s,[4] x [3] sx,. Some simple aalogues of T () are T (,, ), (.7),,, T (.8) T (,, ) ( ) (.9) 0 ad T (.0) (,, ) [ ] [ ]. It is easily verified that s T x,, T x, s,. (.) For we get T( x, s, ) st ( x, s, ) ad T ( x, s, ) xt( x, s, ) st ( x, s, ). (,, ),,,,,, 3, 3 T x s x s xs s s x s xs,. This is This gives the trivial seuece the reaso for excludig. 0 9

10 Propositio. The Chebyshev polyomials of the first id satisfy x s ( x ) s ( ) x 0 0 T ( x, s, ) det ( ) x s ( ) x This is easily see by expadig this determiat with respect to the last colum. Defiitio. The Chebyshev polyomials of the secod id are defied by the recurrece U( xs,, ) ( ) xu ( xs,, ) su ( xs,, ) (.) with iitial values U ( x, s, ) ad U ( x, s, ) The first terms are,[] x,[4] x s,[4]( ) x [4] sx,. Some simple aalogues of (.4) are,, [ ], U 0 (.3) U (,, ). (.4) ad U (,, ) 0 (.5) U (.6) (,, ) [ ]. It is easily verified that U( x, s, ) U s, x,. (.7) 0

11 For we would have U ( x, s, ) ( s) ad U ( x, s, ) 0. Propositio. The Chebyshev polyomials of the secod id satisfy ( x ) s ( ) x s ( ) x 0 0 U( x, s, ) det ( ) x s ( ) x I [3] ad [6] a tilig iterpretatio of the classical Chebyshev polyomials has bee give. This ca easily be exteded to the case. As i the classical case it is easier to begi with polyomials of the secod id. We cosider a rectagle (called board) where the cells of the board are umbered to. As i [3] ad [6] we cosider tiligs with two sorts of suares, say white ad blac suares, ad domioes (which cover two adacet cells of the board). Defiitio.3 To each tilig of a board we assig a weight w i the followig way: Each white suare has i weight x. A blac suare at positio i has weight x ad a domio which covers positios i i, i has weight s. The weight of a tilig is the product of its elemets. The weight of a set of tiligs is the sum of their weights. Each tilig ca be represeted by a word i the letters { abdd,, }. Here a deotes a white suare, b a blac suare ad dd a domio. For example the word abbddaddaab represets the tilig with white suares at positios 7,8. Its weight is, 6, 9,0, blac suares at,3, ad domioes at 4,5 ad x x x s x s x x x s x. Theorem. The weight w( V ) of the set V of all tiligs of a board is wv ( ) U ( x, s, ). Proof This holds for ad. E ach tilig u has oe of the followig forms: u a, u b, u dd. Therefore

12 wv ( ) wu ( ) wu ( ) x wu ( ) x wu ( ) s uv uv uv uv wv ( ) xwv ( ) s which implies Theorem.. Remar If we more geerally cosider the weight w r which coicides with w except that a blac i ( r suare at positio i has weight rx we get i the same way that U ) ( x, s, ) wr( V) satisfies ( r ) ( r (,, ) ) (,, ) ( r U x s r xu x s su ) ( x, s, ) ( r) ( r) with iitial values U0 ( x, s, ) ad U ( x, s, ) ( r) x. I this case we get more geerally m m ( r) ( r) ( r) m m ( r) ( r) m Um( xs,, ) Um( xsu,, ) ( x, s, ) sum ( xsu,, ) ( x, s, ). The secod term occurs whe positios ( mm, ) are covered by a domio ad the first term i the other cases. The same reasoig as above gives Propositio.3 Let usr (,,, ) be the The w weight of all tiligs o {,, } with exactly domioes. r (,,, ) (,,, )( usr u sr rx ) u (,, sr, ) s (.8) with iitial values u (,0, sr, ) ( r)( r) ( rx ), u(,0, s, r) ( r) x ad u(,, s, r) 0 for 0. It is ow easy to verify Theorem. The wr weight u(,, s, r) of the set of all tiligs o {,, } with exactly domioes is usr (,,, ) ( r) ( rsx ) (.9) for 0 ad usr (,,, ) 0 for.

13 Proof The iitial values coicide ad by iductio u (, sr,, )( rx ) u (,, sr, ) s ( r) ( r)( rsx ) ( ) ( ) ( ) r r s x ( r) ( r) s x ( r) ( r) ( r) ( r) s x r ( r) ( r) s x ( r). Here we used the recurrece relatios (.) for the biomial coefficiets. Remar. Formula (.9) is the product of s x ( ) ( r) ( r) r. 0 Ilse Fischer [] has foud a combiatorial reaso for this product represetatio. Let v (,,, x) be the wr weight of all tiligs with domioes ad blac suares. The ad v (,,, x) srx v (,,0,) v (,0,, x). (.0) I order to give a combiatorial iterpretatio of this formula we observe that the weight ca also be obtaied from the followig properties. The fact that the weight of a domio at ii, is i s is euivalet with a) each white suare that appears before this domio cotributes a, b) each blac suare that appears before this domio cotributes a, c) each domio that appears before this domio cotributes d) ad the domio itself cotributes s. 3

14 i The fact that the weight of a blac suare at i is xr is euivalet with e) each white suare that appears before this blac suare cotributes a, f) each blac suare that appears before this blac suare cotributes a, g) each domio that appears before this blac suare cotributes a h) ad the blac suare itself cotributes xr. This ca also be reformulated i the followig way: ) Each blac suare cotributes xr, ) each uordered pair of distict blac suares cotributes a, 3) each white suare before a blac suare cotributes a, 4) each domio cotributes s, 5) each uordered pair of distict domioes cotributes, 6) each white suare before a domio cotributes a, 7) each pair of a domio ad a blac suare, where the order is irrelevat, cotributes a, 8) each domio before a blac suare cotributes a. For b) ad g) is euivalet with 7) ad 8). Now cosider the right-had side of (.0). Observe that v (,0,, x) is determied by ), ) ad 3); v (,,0, xis ) determied by 4), 5) ad 6); ad 7) gives. We first distribute the domioes o the board ad let each uoccupied cell have weight. The we distribute the white ad blac suares o the uoccupied cells. Their weight is v (,0,, x). The total weight of the cofiguratio is v (,,0,) v (,0,, x) if each blac suare before a domio cotributes a. For the 6) is satisfied for the computatio of v (,,0,) sice all suares cotribute a (ad thus behave as white suares i this cotext). Thus the right-had side of (.0) satisfies ) to 7), but istead of 8) we have 8 ): each blac suare before a domio cotributes a. Thus we must reverse the order of the domioes ad blac suares to obtai (.0). A euivalet form is Propositio.4 Let t be a tilig of a board with domioes ad blac suares. Reverse the order of the domioes ad blac suares i t ad obtai a tilig T. Deote by A the tilig obtaied by replacig i T each suare with a colourless suare c with weight ad let B be the tilig obtaied by deletig all domioes of T. The w () t w ( A) w ( B). (.) r r r 4

15 Example Cosider the tilig t abbddaddaab with (,, ) (,,3) weight 8 The T abddddabaab ad A ccddddccccc with wr ( A) s wr ( B) ababaab r x This gives wr () t s x. Theorem. implies for r Theorem s r x 7. ad B ababaab with 0 (.) U( x, s, ) ( ) ( ) s x. For the Chebyshev polyomials of the first id the situatio is somewhat more complicated. Here we get Theorem.4 T (,, ) x s is the weight of the subset of all tiligs of {,, } where the last bloc is either a white suare or a domio. Therefore for 0 T( xs,, ) xu ( xs,, ) su ( xs,, ). (.3) Proof It suffices to prove that the right-had side satisfies the iitial values ad the recurrece (.6). xxu ( xs,, ) su( xs,, ) s xu( xs,, ) su3( xs,, ) xu ( xs,, ) xu ( xs,, ) sxu ( xs,, ) sxu ( xs,, ) sxu xs su xs (,, ) 3(,, ) (,, ) (, s, ) x xu x s su x s xu ( x, s, ) su ( x, s, ) 4 xu( xs,, ) su ( xs,, ). 5

16 Theorem.5 The Chebyshev polyomials of the first id are give by ( ) [ ] (,, ) 0 ( ) ( ) ( )[ ] T x s s x [ ] ( ) s x [ 0mod] s. 0 [ ] (.4) Proof Cosider the subset of all tiligs of a board whose last bloc is ot a blac suare. Let ts (,, ) be the weight of all these tiligs with exactly domioes. The ts (,, ) u (, sx, ) u (, ) s. (.5) We show that ( ) [ ] ts (,, ) sx. ( ) ( ) ( )[ ] (.6) This is true for ad. By iductio we get for ts (,, ) u (, sx, ) u (, ) s ( ) s x ( ) s x ( ) s x ( ) s x [ ] ( ) s x [ ] ad for t( s,, ) u(, sx, ) u(, ) s s s. 6

17 For the polyomial T ( x, s) ca also be iterpreted as the weight of the set T of all tiligs which begi with a domio or with a white suare sice i this case the weights of the words c c ad c c coicide. I the geeral case this is ot true. For example for the set T aa, ab, dd has weight w( T ) x x s T ( x, s, ) x x s. But we have Theorem.6 Proof T( xs,, ) xu ( xs,, ) su ( xs,, ). (.7) It suffices to show that the right-had side satisfies recurrece (.6). xxu( xs,, ) su3( xs,, ) s xu3( xs,, ) su4( xs,, ) xu ( xs,, ) xu ( xs,, ) sxu ( xs,, ) sxu ( xs,, ) sxu x s s U x s 3 3 3(,, ) 4(,, ) (, s, ) su3( xs,, ) x xu x s xu ( xs,, ) su ( xs,, ) xu ( x, s, ) su ( x, s, ). Theorem.6 has the followig tilig iterpretatio: Defie aother weight W such that each white suare has weight x, each blac suare at i i positio i has weight x ad each domio at positio ( i, i) has weight s if i. But a domio at positio (, ) has weight s. If we oi the eds of the board to a circle such that the positio after is this ca also be formulated as: If ( i, i, ) are cosecutive poits the a domio at positio ( i, i) has weight s. The T ( x, s, ) is the weight of all such tiligs which have o blac suare at positio. (Note that o the circle there are o domioes at positio (,). ) I order to fid a aalogue of (.3) let us first cosider this idetity i more detail. (, ) (, ) x x s T x s U x s x s is euivalet with (, ) (, ) x x st( x, s) U ( x, s) x s T x s U x s x s x x s x x s x x s T x s x x s U x s T x s U x s x x s (, ) (, ) (, ) (, ). 7

18 Therefore (.3) is euivalet with both idetities ad T ( xs, ) T( xsx, ) x su ( xs, ) (.8) U ( x, s) T ( x, s) U ( x, s) x. (.9) To prove idetity (.8) observe that for a tilig of a ( ) board which does ot ed with a blac suare either eds with two white suares aa or with a domio ad a white suare dda. The weight w of these tiligs is T ( x, s) x. Or it eds with ba or dd. Their weight is ( x su ) ( xs, ). Idetity (.9) simply meas that a arbitrary tilig either eds with a blac suare which gives the weight U (, ) x s x or does ot ed with a blac suare which gives T ( x, s ). For arbitrary this classificatio of the tiligs implies the idetities T ( xs,, ) xt( xs,, ) ( x su ) ( xs,, ) (.30) ad U( xs,, ) T( xs,, ) xu ( xs,, ). (.3) But there is also aother aalogue of (.8): T ( xs,, ) xt( xs,, ) ( x su ) ( xs,, ). (.3) By (.7) we have T ( xs,, ) xu( xs,, ) su ( xs,, ). Therefore by (.) U( xs,, ) xu ( xs,, ) xu ( xs,, ) su ( xs,, ) xu ( x, s, ) su ( x, s, ) T ( x, s, ). Thus U( xs,, ) T( xs,, ) xu ( xs,, ) (.33) ad (.7) implies (.3). As aalogue of (.8) ad (.9) we ca ow choose the idetities (.3) ad (.3) which we write i the form T ( xs,, ) xt( xs,, ) ( x s) U ( xs,, ) U( xs,, ) T( xs,, ) xu ( xs,, ). (.34) Here deotes the liear operator o the polyomials i s defied by p() s p( s). 8

19 To stress the aalogy with (.3) we itroduce a formal suare root commutes with x ad real or complex umbers ad satisfies (.34) i the form A x s ( ) which A ( x s) ad write T ( xs,, ) AU( xs,, ) ( x A)( T( xs,, ) AU ( xs,, )). (.35) i i Sice ( x A)( x A) ( x A)( x A) usig the biomial theorem (.3) we get as aalogue of (.3) p ( x, A) x A x A x A T ( x, s, ) AU ( x, s, ). (.36) This gives Theorem.7 For the Chebyshev polyomials the followig formulae hold: ad p( x, A) p( x, A) T ( x, s, ) x x s 0 0 (.37) p (, ) (, ) x A p x A (,, ). A 0 0 (.38) U x s x x s Proof This follows from (.3) ad the observatio that A ( x s) ( x s). 0 If we expad x s s x 0 0 we get by comparig coefficiets i (.37) ad (.38) Theorem.8 For the idetities ad ( ) [ ] 0 ( ) ( ) ( )[ ] (.39) 9

20 hold. ( ) ( ) (.40) 0 Remar. It would be ice to fid a combiatorial iterpretatio of these idetities. For we get from (.3) T( x, s) ( x s) U ( x, s) ( s). Sice A does ot commute with polyomials i s we caot deduce a aalogue of this formula from (.36). But we ca istead cosider the matrices A x ( x s x. (.4) We the get Theorem.9 T x s x s U x s s U ( x, s, ) Tx,, (,, ) ( ) (,, ) A A A0. (.4) Proof We must show that T ( xs,, ) ( x su ) ( xs,, ) T(,, ) ( ) xs x su (,, ) xs x ( x s s s U( x, s, ) T x,, x U ( x, s, ) T x,, or euivaletly T ( xs,, ) xt( xs,, ) ( x su ) ( xs,, ), U( xs,, ) T( xs,, ) xu ( xs,, ), U ( x, s, ) T x, s, xu ( x, s, ), T xs,, xt xs,, ( x su ) ( xs,, ). This follows from the recurreces (.30), (.3), (.3) ad (.33). 0

21 If we tae determiats i (.4) we get the desired aalogue of T ( x, s) ( x s) U ( x, s) ( s). Theorem.0 T ( x, s, ) T ( x, s, ) ( x s) U ( x, s, ) U ( x, s, ) ( s). (.43) For example for ( xs, ) (, ) this reduces to T (,, ) ( ) [ ] T (,, ) ( ). I [] may other idetities occur. These follow i a easy maer from the idetities obtaied above. Sice the Chebyshev polyomials satisfy a three-term recurrece they are orthogoal with respect to some liear fuctioals, i.e. LT ( ( xst,, ) m( xs,, )) 0 ad MU ( ( xsu,, ) m( xs,, )) 0 for m. These liear fuctioals are uiuely determied by LT (,, ) xs [ 0] ad M U (,, ) x s [ 0]. These liear fuctioals are closely related. From (.30) we get T ( xs,, ) xt( xs,, ) ( x su ) ( xs,, ). T ( x, s, ) st ( x, s, ) By (.6) we have xt ( x, s, ) ad therefore we obtai T ( xs,, ) st ( xs,, ) ( )( x su ) ( xs,, ). (.44) If we apply the liear fuctioal L to this idetity we deduce that x ( ) L U( x, s, ) [ 0] M U( x, s, ). s (.45) By liearity we obtai for all polyomials p( x ). x ( L ) px ( ) Mpx ( ) s (.46)

22 As aalogue of (.4) we get LT if 0 ( s) if 0 s This follows by applyig L to (.6) which gives Lx T Lx T (.47) ad therefore L x T ( )( ) ( ) ( s). Now observe that LT L xt ( ) ( ). Of special iterest are the momets of these liear fuctioals, i.e. the values Lx ( ) ad M ( x ). To fid these values it suffices to fid the uiuely determied represetatio of as a liear combiatio of the Chebyshev polyomials. These have bee calculated i [] for the correspodig moic polyomials. Therefore I oly state the results i the preset otatio: x For the Chebyshev polyomials of the first id we have 0 T ( x, s, ) x ( [ ])( s). ( ) ( )( ) ( ) (.48) This gives as aalogue of (.5) L x ( s) ( ) ad Lx ( ) 0. s For the moic polyomials we get the three-term recurrece with s ( ) 0, t(0) ad s t ( ). ( )( ) (.49)

23 For the Chebyshev polyomials of the secod id the correspodig formulae are as aalogue of (.7) ad MU s ( ) ( ) (.50) ad therefore 0 x ( s) U( x, s, ) ( ) ( ) (.5) M x ( ) ( s) [ ] ( ) ad M( x ) 0. Of course (.5) also follows directly from (.49) ad (.46). (.5) The parameters for the three-term recurrece of the moic polyomials are s ( ) 0 ad s t ( ). ( )( ) Remar.3 The Chebyshev polyomials have also appeared, partly implicitly ad without recogizig them as aalogues of the Chebyshev polyomials, i [6], [7] ad [3] i the course of computig Hael determiats of ( a; ), ( ab ; ) which are the momets of the little Jacobi polyomials p ( xab ;, ) (cf. [4]). Note that Lx ( ; ) ( ) ( s). 4 ( ; ) M x ( ; ) ( ) ( s) ( ; ) ad 3. Expasios of -Chebyshev polyomials with odd idex by those with eve idex The Chebyshev polyomials T(, s, ), T (, s, ), U (,, ) s ad U (,, ) s are polyomials i s of degree. Therefore there exist uiue represetatios ad T (, s, ) a(,, ) T (, s, ) (3.) 0 3

24 U (, s, ) b(,, ) U (, s, ). (3.) 0 To obtai these represetatios we eed aalogues of the taget ad Geocchi umbers. The taget umbers t ( ) are well-ow obects defied by the geeratig fuctio ez ( ) e( z) ( ) t ( ) z ez ( ) e( z) []!. (3.3) 0 Theorem 3. T ( x, s, ) ( ) t ( ) x T( x, s, ). 0 (3.4) Proof I (.37) we have see that 3 T (, s, ) ( s)( s) ( s). 0 This implies that satisfies T (, s, ) T( z, s, ) z (3.5) [ ]! z T( z, s, ) ( s)( s) ( s). (3.6) e( z) [ ]! Therefore e( zt ) ( zs,, ) ezt ( ) ( zs,, ) ad ez ( ) e( z) T( zs,, ) T( zs,, ) ez ( ) e( z) T( zs,, ) T( zs,, ) or T (, s, ) z 0 [ ]! ez ez t T (, s, ) ez ( ) e( z) 0 []! 0 [ ]! z ( ) ( ) ( ) ( ) z. (3.7) Note that the left-had side does ot deped o s. If we choose s 0 we get that 0 ; z [ ]! ez ( ) e( z). ; ez ( ) e( z) z [ ]! (3.8) 4

25 (3.7) implies T (, s, ) ( ) t ( ) T (, s, ) z z z []! []! [ ]! which gives by comparig coefficiets T (, s, ) ( ) t ( ) T(, s, ) 0 (3.9) ad therefore also (3.4). To obtai the expasio (3.) we defie Geocchi umbers G ( ) by the geeratig fuctio ez ( ) e( z) ( ) G( ) ; z z. (3.0) ez ( ) e( z) [ ]! 0 This implies that t ( ). G ( ) ; [ ] (3.) (Observe that this aalogue of the Geocchi umbers does ot coicide with the Geocchi umbers itroduced by J. Zeg ad J. Zhou which have bee studied i [9]). The first terms of the seuece G ( ) are G( ), G4( ), 3 ( )( )( ) G6( ), 4 5 ( )( ) ( ) ( ) G8( ) ( )( )( ) Theorem 3. U ( x, s, ) ; ( ) G( ) x U( x, s, ). 0 [ ] (3.) 5

26 Proof 3 I (.38) we have see that U(, s, ) ( s)( s) ( s). 0 By comparig coefficiets this is euivalet with z 3 U (, s, ) ( s)( s) ( s) z. (3.3) e( z) []! [ ]! 0 Let ow We the get U z s U (, s, ) (3.4) [ ]! (,, ) z. z 3 e( zu ) ( zs,, ) ( s)( s) ( s) ezu ( ) ( zs,, ). 0 [ ]! This implies ( ez ( ) e( z))( Uzs (,, ) U( zs,, )) ezu ( ) ( zs,, ) e( zuzs ) (,, ) ezuzs ( ) (,, ) e( zu ) ( zs,, ) ezu ( ) ( zs,, ) e( zu ) ( zs,, ) ( ez ( ) e( z))( U( zs,, ) U( zs,, )). U (, s, ) Sice U( zs,, ) U( zs,, ) z ad [ ]! U(, s, ) U( zs,, ) U( zs,, ) z 0 [ ]! we see that U (, s, ) z [ ]! ez ( ) e( z) U(, s, ) ez ( ) e( z) 0 z [ ]!. (3.5) Agai the left-had side does ot deped o s. So we ca e.g. choose s 0 ad get that ( ; ) z [ ]! ez ( ) e( z). (3.6) ( ; ) z ez ( ) e( z) [ ]! If we write (3.5) i the form 0 U (, s, ) e( z) e( z) U (, s, ) z z z [ ]! e( z) e( z) []! 0 ad compare coefficiets we get 6

27 U (, s, ) ; ( ) G ( U ) (, s, ). 0 [ ] This immediately implies Theorem 3.. The Geocchi umbers itroduced i (3.0) have some iterestig properties which do ot immediately follow from their relatio with the taget umbers (3.). ( ; ) Sice the left-had side of (3.0) ad [ ]! are ivariat uder we see that G ( ). G (3.7) Lemma 3. The Chebyshev polyomials U (, s, ) satisfy the idetity ( ) ( ) U (, s, ) 0. 0 (3.8) More geerally for all o-egative itegers m holds. m m m m 0 m (3.9) ( ) ( ) U (, s, ) s U (, s, ) Proof Let m (,,, ) ( ) ( ) m(,, ). 0 m (3.0) W m s U s We wat to show that m W(, m, s, ) s U (, s, ). (3.) m We prove this idetity with iductio. For 0 it is the trivial idetity Um (, s, ) Um (, s, ). m m For it reduces to Um (, s, ) ( ) Um(, s, ) sum (, s, ). By defiitio of the polyomials this is true for all o-egative m. 7

28 I geeral we have m W(, m, s, ) W(, m, s, ) ( ) W(, m, s, ). (3.) Observig that m m ( ) ( ) m m we get W m s W m s m (,,, ) ( ) (,,, ) m m m ( ) ( ) U (, s, ) ( ) ( ) ( ) U (, s, ) m m 0 m 0 m m m m ( ) ( ) U (, s, ) ( ) ( ) ( ) U (, s, ) m m 0 m m U s U m m m (,, ) ( ) ( ) (, s, ) ( ) ( ) m m m m m ( )( ) ( ) U (, s, ) m m m ( ) ( ) U (, s, ) W(, m, s, ). m 0 m By iductio (3.) implies W m s W m s W m s m (,,, ) (,,, ) ( ) (,,, ) ( ) m ( ) m m m(,, ) ( ) m(,, ) ( ) m m m m m m s U s s U s s U (, s, ) ( ) U (, s, ) s U (, s, ). For m 0 we get (3.8). A easy coseuece is a aalogue of the Seidel formula for the Geocchi umbers which gives a easy way to calculate the Geocchi umbers ad shows that ( ; ) G ( ) [ ] is a polyomial with iteger coefficiets. Theorem 3.3 (-Seidel formula) 8 ; ( ) G ( ) [ ]. 0 ; (3.3)

29 Proof Sice the set of polyomials U (, s, ) is a basis for the vector space of polyomials i 0 U (, s, ) [ 0]. s we ca defie a liear fuctioal by By (3.) this implies U (, s, ) ( ) ; G ( ). If we apply this to (3.8) we get for 0 ( ) ( ) U (, s, ) ( ) ( ) ( U (, s, )) 0 0 ( ). ( ) ( )( ) G 0 Dividig by ; we get (3.3). Refereces [] Waleed A. Al-Salam ad Mourad E.H. Ismail, Orthogoal polyomials associated with the Rogers-Ramaua cotiued fractio, Pacific J. Math. 04 (983), [] Natig Ataishiyev, Pedro Fraco, Decio Levi ad Orlado Ragisco, O Fourier itegral trasforms for Fiboacci ad Lucas polyomials, J.Phys.A:Math.Theor.,Vol.45,No.9,Art.No.9506, pages, 0. [3] Arthur T. Beami ad Daiel Walto, Coutig o Chebyshev polyomials, Math. Mag. 8 (), (009), 7-6, [4] T. S. Chihara, A itroductio to orthogoal polyomials, Gordo ad Breach 978 [5] Joha Cigler, Elemetare -Idetitäte, Sém. Lotharigie Comb. B05a (98) [6] Joha Cigler, A simple approach to some Hael determiats, arxiv: [7] Joha Cigler, How to guess ad prove explicit formulas for some Hael determiats, 00, [8] Joha Cigler, Lucas polyomials ad associated Rogers-Ramaua idetities, arxiv: [9] Joha Cigler, -Fiboacci polyomials ad -Geocchi umbers, arxiv: [0] Joha Cigler, Some beautiful -aalogues of Fiboacci ad Lucas polyomials, arxiv: [] Joha Cigler, A simple approach to -Chebyshev polyomials, arxiv: [] Ilse Fischer, Persoal commuicatio,

30 [3] Masao Ishiawa, Hiroyui Tagawa ad Jiag Zeg, A -aalogue of Catala Hael determiats, arxiv: [4] R. Koeoe, P.A. Lesy ad R.F. Swarttouw, Hypergeometric orthogoal polyomials ad their - aalogues, Spriger Moographs i Mathematics 00 [5] Theodore J. Rivli, The Chebyshev polyomials, Wiley 974 [6] Daiel Walto, A tilig approach to Chebyshev polyomials, Thesis