q-Classical polynomials and the q-Askey and Nikiforov-Uvarov tableaus

q  Classial p olynomials and he q  Askey and Niki o o -U a o Tableaus R.  Al a ez-No da se a;b 1 and J. C. Medem a 2 a Depa amen o de Analisis Ma ema io. Uni e sidad de Se illa. Ap do. 1160, E-41080 Se illa, Spain b Ins i u o Ca los I de Fsia Teo ia y Compu aional, Uni e sidad de G anada, E-18071 G anada, Spain Janua y 8, 2000 Abs a In his pap e we on inue he s udy o he q  lassial (dis e e) p olynomials (in he Hahn's sense) s a ed in [18℄. He e we will ompa e ou sheme wi h he well known q  Askey Sheme and he Niki o o -U a o Tableau. Also, new amilies o q  p olynomials a e in o dued. In o du ion The so-alled q  p olynomials ons i u e a e y imp o an and in e es ing se o sp eial un ions and mo e sp eially o o hogonal p olynomials. They app ea in se e al b anhes o he na u al sienes, e.g., on inued a ions, Eule ian se ies, he a un ions, ellip i un ions,...; see [3 , 9℄, quan um g oups and algeb as [14, 15, 25 ℄, among o he s (see also [10 , 20 ℄). They ha e b een in ensi ely s udied in he las yea s by se e al p eople (see e.g. [13℄) using se e al o ols. One o hem is he one e iewed in [13 ℄ whih is based on he basi hyp e geome i se ies [10 ℄ and was de elop ed mainly by he Ame ian Sho ol s a ing by he wo ks o And ews and Askey (see e.g. [4℄, he li e a u e on his me ho d is so as ha we a e no able o inlude i he e, a e y omple e lis is gi en in [13℄) and lead o he so-alled q  Askey Tableau o hyp e geome i p olynomials [13℄. In o he di e ion, he Russian ( o me So ie ) sho ol, s a ing om he wo ks by Niki o o and U a o [21 ℄ and u he de elop ed by A akishiye and Suslo (see e.g. [5 , 6, 20, 23 , 24℄ and e e enes on ained he ein), ha e onside ed he die ene analog in non-uni o m la ies o he hyp e geome i die en ial equa ion [22℄, om whe e he hyp e geome i ep esen a ion o he q  p olynomials ollows in a e y simple way [5, 23℄. This shema leads o he Niki o o -U a o ableau [20 , 23℄ o he p olynomial solu ions o he die ene hyp e geome i equa ion on non- uni o m la ies. A sp eial men ion dese es he pap e by A akishie and Suslo [6 ℄ whe e a die ene analog o he well known me ho d o unde e mina ed o eÆien s ha e b een de elop ed o he hyp e geome i equa ion on non-uni o m la ies and also gi e a lassia ion simila o he Niki o o and U a o 1991 one bu o he q  sp eial un ions (no only o he p olynomials solu ions). Ou main aims he e a e wo: o on inue he s udy s a ed in [18℄ using he algeb ai heo y de elop ed by Ma oni [16℄ and o lassi y he q  lassial p olynomials and ompa e wi h he q  Askey and Niki o o & U a o Tableaus. In a , in [18 ℄ we ha e p o en se e al ha a e iza ion o he q  lassial p olynomials as well as a e y simple ompu a ional algo i hm o nding hei main ha a e is is (e.g. he o eÆien s o he h ee- e m eu en ela ion, s u u e ela ion o Al-Salam Chiha a, e ). Going u he , we will gi e he e a e y na u al" lassia ion o he q  lassial p olynomials in o dued by Hahn in his pap e [11℄, i.e., we will lassi y all o hogonal p olynomial sequenes suh ha hei q  die enes, dened by  ( x ) = ( q x )  ( x ) ( q  1) x a e o hogonal in he widesp ead sene: he q  Hahn Tableau (a  s s ep on his in he ame wo k o he q  Askey ableau was done in [15℄). No ie ha he a o esaid p olynomials a e ins anes o he 1 Phone: +34 954 55 7997, Fax: 954 55 7972. E-mail: ania.es 2 Phone: +34 954 55 7997, Fax: 954 55 7972. E-mail: jmedemia.es 1 2 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus q  p olynomials on he linea exp onen ial la ie x ( s ) =  1 q s . Fo se e al su eys on his la ie and hei o esp onding p olynomials see [2, 4, 5 , 8 , 10 , 13, 20, 24℄. (see also se ion 3.2 om b elow). Fu he mo e, we will ompa e ou lassia ion ( q  Hahn Tableau) wi h he a o esaid wo Shemas. F om his ompa a ion we nd ha he e a e missing amilies in he q  Askey Shema (one o hem is a non-p osi i e deni e amily) and using he esul s o [18 ℄ we s udy hem wi h de ails. Also he o esp ondene o his q  Hahn Shema and he Niki o o & U a o one will b e s ablished. In suh a way a omple e o esp ondene b e ween he q  lassial amilies o he q  Askey and Niki o o & U a o Tableaus o exp onen ial linea la ies will b e shown. The s u u e o he pap e is as ollows. In Se ion 1 we in o due some no a ions and deni ions use ul o he nex ones. In Se ion 2, he q  weigh un ions a e in o dued and ompu ed o all q  lassial amilies. This will allow o lassi y all o hogonal p olynomial amilies o he q  Hahn ableau. Finally, in Se ion 3, se e al applia ions a e onside ed: he lassia ion o he q  lassial p olynomials ( q  Hahn Tableau), he in eg al ep esen a ion o he o hogonali y, he hyp e geome i ep esen a ion o hese q  lassial p olynomials as well as he de ailed s udy o wo new amilies o q  p olynomials. 1 P elimina ies In his se ion we will gi e a b ie su ey o he op e a ional alulus and some basi onep s and esul s needed o he es o he wo k. Le P b e he linea spae o p olynomial un ions in C wi h omplex o eÆien s and P  b e i s algeb ai dual spae, i.e., P  is he linea spae o all linea applia ions u : P ! C . In he ollowing we will e e o he elemen s o P  as un ionals and we will deno e hem wi h b old le e s ( u ; ;::: ). Sine he elemen s o P  a e linea un ionals, i is p ossible o de e mine hem om hei a ions on a gi en basis ( B n ) n  0 o P , e.g. he anonial basis o P , ( x n ) n  0 . In gene al, we will ep esen he a ion o a un ional o e a p olynomial by o mula h u ;  i ; u 2 P  ;  2 P , and he e o e a un ional is omple ely de e mined by a sequene o omplex numb e s h u ; x n i = u n , n  0, he so-alled momen s o he un ional. Deni ion 1.1 Le ( P n ) n  0 be a basis sequene o P . We say ha ( P n ) n  0 is an o hogonal polynomial sequene (OPS in sho ), i and only i he e exis s a un ional u 2 P  suh ha h u ; P m P n i = k n Æ mn , k n 6 = 0 ; n  0 , whe e Æ mn is he K oneke del a. I k n > 0 o al l n  0 , we say ha ( P n ) n  0 is a posi i e deni e OPS. Deni ion 1.2 Le u 2 P  be a un ional. We say ha u is a quasi-deni e un ional i and only i he e exis s a polynomial sequene ( P n ) n  0 , whih is o hogonal wi h espe o u . I ( P n ) n  0 is posi i e deni e, we say ha u is a posi i e deni e un ional. Deni ion 1.3 Gi en a polynomial sequene ( P n ) n  0 , we say ha ( P n ) n  0 is a moni o hogonal polynomial sequene (MOPS in sho ) wi h espe o u , and we deno e i by ( P n ) n  0 = mops u i and only i P n ( x ) = x n + lowe deg ee e ms and h u ; P m P n i = k n Æ nm ; k n 6 = 0 ; n  0 . Also he nex heo em will b e use ul Theo em 1.1 ( Fa a d Theo em [7℄) Le ( P n ) n  0 be a moni polynomial basis sequene. Then, ( P n ) n  0 is an MOPS i and only i he e exis wo sequenes o omplex numbe s ( d n ) n  0 and ( g n ) n  1 , suh ha g n 6 = 0 , n  1 and xP n = P n +1 + d n P n + g n P n  1 ; P  1 = 0 ; P 0 = 1 ; n  0 ; (1.1) whe e P  1 ( x )  0 and P 0 ( x )  1 . Mo eo e , he un ional u wi h espe o whih he polynomials ( P n ) n  0 a e o hogonal is posi i e deni e i and only i ( d n ) n  0 is a eal sequene and g n > 0 o al l n  1 . R.  Al a ez-Noda se and J. C. Medem 3 In he ollowing, we will use he no a ion: Deni ion 1.4 Le  2 P and a 2 C , a 6 = 0 . We al l he ope a o H a : P ! P , H a  ( x ) =  ( ax ) , a dila ion o a io a 2 C n 0 g . This op e a o is linea on P and sa ises H a (   ) = H a   H a  . Also no ie ha o any omplex numb e a 6 = 0, H a  H a  1 = I, whe e I is he iden i y op e a o on P , i.e., o all a 6 = 0, H a has an in e se op e a o . In he ollowing we will omi any e e ene o q in he op e a o s H q and hei in e se H q  1 . So, H := H q , H  1 := H q  1 . Nex , we will dene he so alled q  de i a i e op e a o [11 ℄. We will supp ose also ha j q j 6 = 1 (al hough i is p ossible o weak his ondi ion). Deni ion 1.5 Le  2 P and q 2 C n 0 g , j q j 6 = 1 . The q  de i a i e ope a o  , is he ope a o  : P ! P , dened by   = H    H x  x = H    ( q  1) x : The q  1  de i a i e ope a o  ? , is he ope a o  ? : P ! P dened by  ?  = H  1    H  1 x  x = H  1    ( q  1  1) x : In his way,   and  ?  wil l deno e he q  de i a i e and q  1  de i a i e o  , espe i ely. The ab o e wo op e a o s  and  ? a e linea op e a o s on P , and  x n = H x n  x n ( q  1) x = ( q n  1) x n ( q  1) x = [ n ℄ x n  1 ; n > 0 ; 1 = 0 ; (1.2) i.e.,   2 P . He e [ n ℄ ; n 2 N , deno es he basi q  numb e n dened by [ n ℄ = q n  1 q  1 = 1 + q + ::: + q n  1 ; n > 0 ; [0℄ = 0 : (1.3) Also he q  1 numb e s [ n ℄ ? , dened by [ n ℄ ? = q  n  1 q  1  1 = q 1  n [ n ℄ will b e used. No ie ha  ? is no he in e se o . In a hey a e ela ed by H  ? =  ; H  1 = ? . The q  de i a i e sa ises he p o du ule (   ) =    + H     = H     +    . Deni ion 1.6 Le ! a de i able un ion a x = 0 suh ha 8 a 2 dom ! , aq 2 dom ! . Then, we wil l dene he q  de i a i e o ! by he exp ession  ! = H !  ! H x  x = H !  ! ( q  1) x ; x 6 = 0 ;  ! (0) = ! 0 (0) : (1.4) Deni ion 1.7 Le u 2 P  and  2 P . We dene he a ion o a dila ion H a and he q  de i a i e  on P  by he exp essions H a : P  ! P  , h H a u ;  i = h u ; H a  i ,  : P  ! P  , h  u ;  i = h u ;   i , espe i ely. Deni ion 1.8 Le u 2 P  and  2 P . We dene a polynomial modia ion o a un ional u , he un ional  u , h  u ;  i = h u ;   i ; 8  2 P . No ie ha we use he same no a ion o he op e a o s on P and P  . Whene e i is no sp eied on whih linea spae an op e a o a s, i will b e unde s o o d ha i a s on he p olynomial spae P . Deni ion 1.9 Le u 2 P  be a quasi-deni e un ional and ( P n ) n  0 = mops( u ) . We say ha u o ( P n ) n  0 a e q  lassi un ional o MOPS, espe i ely, i and only i he sequene ( P n +1 ) n  0 is also o hogonal. 4 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus No ie ha in he Hahn deni ion [11℄ q is a eal pa ame e and he e, in gene al, q 2 C n 0 g , j q j 6 = 1. In he ollowing ( Q n ) n  0 will deno e he sequene o moni q  de i a i es o ( P n ) n  0 , i.e., Q n = 1 [ n +1℄ P n +1 , o all n  0. Theo em 1.2 (Medem e al. [17, 18 ℄) Le u 2 P  be a quasi-deni e un ional. and ( P n ) n  0 = mops( u ) . Then, he ol lowing s a emen s a e equi alen : (a) u and ( P n ) n  0 a e, espe i ely, a q  lassial un ional and a q  lassial MOPS. (b) The e exis s a pai o polynomials  and , deg   2 , deg = 1 , suh ha (  u ) = u : (1.5) () ( P n ) n  0 sa ises he q  SL die ene equa ion   ? P n +  ? P n = b  n P n ; n  0 ; (1.6) i.e., P n a e he eigen un ions o he S u m-Liou il le ope a o   ? +  ? o esponding o he eigen alues b  n . Mo eo e , i  ( x ) = b ax 2 +  ax + _ a; ( x ) = b bx +  b; b b 6 = 0 ; (1.7) hen, he quasi-deni eness o u implies [ n ℄ b a + b b 6 = 0 and he ol lowing equi alenes hold [ n ℄ b a + b b 6 = 0 ; n  0 () b  n 6 = b  m ; 8 n; m  1 ; n 6 = m () b  n 6 = 0 ; 8 n  1 : Theo em 1.3 Le u 2 P  , be a quasi-deni e un ional, ( P n ) n  0 = mops u and Q ( k ) n = 1 [ n +1℄ ( k )  k P n + k , whe e [ n + 1℄ ( k )  [ n + 1℄[ n + 2℄ ::: [ n + k  1℄ . The ol lowing s a emen s a e equi alen : (a) ( P n ) n  0 is q  lassial, (b) ( Q ( k ) n ) n  0 is q  lassial, k  1 . Mo eo e , i u sa ises he equa ion (  u ) = u , deg   2 and deg = 1 , hen ( Q ( k ) n ) is o hogonal wi h espe o ( k ) = H ( k )   u , H ( k ) = Q k i =1 H i  1  , and i sa ises (  ( k ) ( k ) ) = ( k ) ( k ) ; deg  ( k )  2 deg ( k ) = 1 ; whe e  ( k ) = H k  and ( k ) = +  P k  1 i =0 H i  , and hey a e he polynomial solu ions o he q  SL equa ion SL ( k ) Q ( k ) n =  ( k )  ? Q ( k ) n + ( k )  ? Q ( k ) n = b  ( k ) n Q ( k ) n ; (1.8) whe e he polynomials  ( k ) and ( k ) and he eigen alues b  ( k ) n a e  ( k ) = q 2 k b ax 2 + q k ax +  a ; ( k ) = ([2 k ℄ b a + b b ) x + ([ k ℄ a + b ) ; b  ( k ) n = [ n ℄ ? ([2 k + n  1℄ b a + b b ) : (1.9) Fu he mo e, in [17 , 18℄ he ollowing esul was p o en: Theo em 1.4 Le u 2 P  , be a quasi-deni e un ional, ( P n ) n  0 = mops u , ;  ? ; 2 P , suh ha  ? = q  1  + ( q  1  1) x , deg   2 , deg  ?  2 and deg = 1 . Then, he ol lowing s a emen s a e equi alen (a) u and ( P n ) n  0 = mops u a e q  lassial and (  u ) = u , (b) u and ( P n ) n  0 = mops u a e q  1  lassial and  ? (  ? u ) = u . R.  Al a ez-Noda se and J. C. Medem 5 () The e exis a polynomial  2 P , deg   2 and h ee sequenes o omplex numbe s a n ; b n ;  n ,  n 6 = 0 , suh ha   P n = a n P n +1 + b n P n +  n P n  1 ; n  1 ; (1.10) (d) he e exis a omplex numbe s e n ; h n , suh ha P n = Q n + e n Q n  1 + h n Q n  2 ; n  2 : (1.11) (e) The e exis a polynomial  2 P , deg   2 and a sequene o omplex numbe s n , n 6 = 0 , n  1 suh ha P n u = n  n (H ( n )   u ) ; H ( n )  = n Y i =1 H i  1  ; n = q ( n 2 ) n Y i =1  [2 n  i  1℄ b a + b b   1 ; n  1 : (1.12) 2 The q  weigh un ion ! 2.1 Deni ion and  s p op e ies In his se ion we will onside he so-alled weigh un ions o q  lassial p olynomials. The nex p op osi ion an b e p o en s aigh o wa d (see e.g. [12 ℄). P op osi ion 2.1 Le ! a un ion suh ha i a 2 dom ! , aq  1 2 dom ! and ha sa ises he die ene equa ion  ? ( ! ) = q ! () ! = q H(  ? ! ) ; ; 2 P ;  ? = q  1  + ( q  1  1) x : (2.1) Then, he ol lowing wo equa ions a e equi alen   ? P n +  ? P n = b  n P n ; ()  ? ( !  P n ) = q b  n ! P n ; n  1 : (2.2) The ab o e p op osi ion allows us o gene alize he lassial p o edu e o he q  ase o ob aining almos all he ha a e is is o he MOPS. The equa ion (2.1) is usually alled he q  Pea son equa ion and i s solu ion ! is known as he q  weigh un ion and i allows o ew i e he S u m- Liuo ille equa ion (1.6) in i s sel -adjoin o m (2.2). Mo eo e , he weigh un ion ! allow us o ob ain he s anda d" q  Ro d igues o mula and also jus i y he q  in eg al ep esen a ion o he o hogonali y ela ion. In suh a way i is na u al o gi e he ollowing Deni ion 2.1 Le u 2 P  , be a quasi-deni e un ional sa is ying he dis ibu ional equa ion (1.5), whe e ; 2 P , deg   2 , deg = 1 and ( P n ) n  0 = mops u . We say ha ! is he q  weigh un ion assoia ed o u ( espe i ely o ( P n ) n  0 ) i ! sa ises he equa ion (2.1)  ? ( ! ) = q ! . The las deni ion allows us o ew i e he q  SL equa ion (1.8) in i s sel -adjoin o m. In a , an s aigh o wa d alula ions show ha , i ! ( k ) sa ises he q  Pea son equa ion  ? (  ( k ) ! ( k ) ) = q ( k ) ! ( k ) ; (2.3) whe e  ( k ) and ( k ) a e gi en in (1.9), hen (1.8) an b e ew i en in i s sel -adjoin o m  ? (  ( k ) ! ( k )  Q ( k ) n ) = q b  ( k ) n ! ( k ) Q ( k ) n ; n  1 ; k = 0 ; 1 ;::: ;n: (2.4) P op osi ion 2.2 Le ! he solu ion o (2.1) and ! ( k ) he solu ion o (2.3). Then, ! ( k ) =  ( n  1) ! ( n  1) =  = H ( n )   ! ; ! (0)  ! : (2.5) 6 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus P o o : We s a om he q  Pea son equa ion (2.3) and ew i e i in i s equi alen o m  ( k ) ! ( k ) = q H[  ( k ) ℄ ? H ! ( k ) , whe e [  ( k ) ℄ ? = q  1  ( k ) + ( q  1  1) x ( k ) =  ? , o all k 2 N . Thus, by subs i u ing ! ( k ) = H ( n )   ! in  ( k ) ! ( k ) = q H  ? H ! ( k ) , we nd  ( k ) ! ( k ) = q H[  ( k ) ℄ ? H ! ( k ) () H k  (  H   H k  1   ! ) = q H  ? H   H k  H ! () ! = q H  ? H ! ()  ? ( ! ) = q ! ; om whe e he p op osi ion ollows.  Rema k 2.1 No ie ha he polynomials (  ( k ) ) ? and (  ? ) ( k ) a e e y die en . In a , he  s one oge he wi h ( k ) a e he o esponding polynomials ha appea in he q  1  dis ibu ional equa ion sa ised by he un ional ( k ) , i.e., he un ional wi h espe o whih he k  h moni de i a i es Q ( k ) n a e o hogonal, (see P oposi ion 1.4) (  ( k ) ( k ) ) = ( k ) ( k ) ()  ? (  ( k ) ) ? ( k ) = ( k ) ( k ) ; (  ( k ) ) ? =  ? ; 8 k 2 N ; whe eas he seond one join wi h ( ? ) ( k ) a e he polynomial oeÆien s o he q  1  SL equa ion ( ? ) ( k )  ?  Q ? ( k ) n ( ? ) ( k ) = ( b  ? ) ( k ) n Q ? ( k ) n , o he n  h q  1  de i a i e Q ? ( k ) n o he polynomials P n , Q ? ( k ) n = 1 [ n +1℄ ? ( k ) [ ? ℄ n P n + k o he q  1  dis ibu ional equa ion sa ised by he un ional ? ( k ) ,  ? [(  ? ) ( k ) ? ( k ) ℄ = ( ? ) ( k ) ? ( k ) ; (  ? ) ( k ) = H  k  ? ; 8 k 2 N ; i.e., he un ional wi h espe o whih he k  h moni de i a i es Q ? ( k ) n a e o hogonal. 2.2 Compu a ion o he q  weigh un ions This se ion is de o ed o ob ain he q  weigh un ion asso ia ed o all q  lassial un ionals, i.e., he quasi-deni e un ionals o esp onding o he MOPS in he widesp ead sense  u ; P 2 n  6 = 0, o all n  0. In a , Theo em 2.1 and 2.2 will gi e, in a e y na u al way, he key o he lassia ion o all q  lassial o hogonal p olynomials. In he ollowing we onside he ase when j q j < 1 ( j q  1 j > 1). Also we will use he s anda d no a ion ( a ; q ) n = (1  a )(1  aq )  (1  aq n  1 ) o n  1, ( a ; q ) 0  1 o he q  analogue o he Po hamme symb ol, and ( a ; q ) 1 = Q 1 n =0 (1  aq n ), o he absolu ely on e gen inni e p o du o j q j < 1. Fi s o all, we will ew i e he q  Pea son equa ion (2.1)  ? ( ! ) = q ! () ! = q H  ? H ! ()  ? ! = q  1 H  1  H  1 ! ; (2.6) and sol e he esul ing equa ion by he eu en p o edu e shown in gu e 1. Figu e 1. Reu en shema using he q  dila ion. w = H n w  q H  ?   H q H  ?   :::  H n  1 q H  ?  |{z } H ( n ) q H  ?   = Q n  1 k =0 q  ? ( q k +1 x )  ( q k x )  H 2  H 2 w = H 2 ( q H  ? )H 3 w ::: :::             1      R          1   R H  H w = H( q H  ? )H 2 w ? H w = H 2 w q H  ?  H q H  ?  w = q H  ? H w - ? H w = H w q H  ?  R.  Al a ez-Noda se and J. C. Medem 7 In he ase when ! is on inuous a 0 and ! (0) 6 = 0, aking he limi n ! 1 , we nd, sine lim n !1 H n w = lim n !1 w ( q n x ) = w (0), ! = ! (0) lim H ( 1 ) q H  ?  = ! (0) lim n !1 H ( n ) q H  ?  = ! (0) 1 Y n =0 q H  ?  : (2.7) The nex s ep is o ob ain an explii exp ession o he p o du H ( 1 ) q H  ?  . Fo doing ha we need a lemma whih is in e es ing in i s own igh . Lemma 2.1 I  is an n  h deg ee polynomial wi h an independen e m  (0) = 1 , and ze os a i 2 C n 0 g , i = 1 ; 2 ;::: ;n , hen H ( 1 )  = ( a  1 1 x ; q ) 1 ( a  1 2 x ; q ) 1  ( a  1 n x ; q ) 1 := ( a  1 1 x; a  1 2 x;  ; a  1 n x ; q ) 1 ; is an en i e un ion o x wi h ze os a a i q  k , i = 1 ; 2 ;::: ;n and k  0 . Fu he mo e, i  = is a a ional un ion suh ha  (0) =  (0) 6 = 0 and wi h non- anishing ze os o i s nume a o and denomina o , hen, H ( 1 )   = ( a  1 1 x ; q ) 1 ( a  1 2 x ; q ) 1  ( a  1 n x ; q ) 1 ( b  1 1 x ; q ) 1 ( b  1 2 x ; q ) 1  ( b  1 m x ; q ) 1 = ( a  1 1 x; a  1 2 x;  ; a  1 n x ; q ) 1 ( b  1 1 x; b  1 2 x;  ; b  1 m x ; q ) 1 ; i is a me omo phi un ion wi h ze os a a i q  k , i = 1 ; 2 ;::: ;n and k  0 and poles a b j q  l , j = 1 ; 2 ;::: ;m and l  0 , whe e a i 2 C , i = 1 ; 2 ;::: ;n and b k 2 C , k = 1 ; 2 ;::: ;m , a e he ze os o he nume a o and denomina o o  = , espe i ely. P o o : The p o o is based on he a ha , i  is a p olynomial o deg ee n wi h non anishing ze os and  (0) = 1, hen i admi s he a o iza ion  = A ( x  a 1 )( x  a 2 )  ( x  a n ) = (  1) n Aa 1 a 2  a n |{z }  (0)=1 (1  a  1 1 x )(1  a  1 2 x )  (1  a  1 n x ) : Then, H ( k )  = ( a  1 1 x; a  1 2 x;  ; a  1 n x ; q ) k and so, H ( 1 )  = ( a  1 1 x; a  1 2 x;  ; a  1 n x ; q ) 1 . This un- ion is an en i e un ion due o he Weie s ass Theo em (see e.g. [1, x 4.3℄). The p o o o he seond s a emen is analogous and he un ion H ( 1 )   is me omo phi b eause is a quo ien o wo en i e un ions (see e.g. [1, x 4.3℄).  Now, i  (0) 6 = 0, he ab o e lemma leads us o he ollowing well known esul [11℄ Theo em 2.1 Le ( P n ) n  0 = mops u sa is ying he q  S u m-Liou il le equa ion (1.6). I we deno e by a 1 and a 2 he ze os o  and by a ? 1 and a ? 2 he ze os o  ? (see P oposi ion 1.4), and al l hey a e die en om 0, hen he ol lowing exp essions o he q  weigh un ions ! hold   ? q  weigh un ion ! ( x ) b a ? ( x  a ? 1 )( x  a ? 2 ) , b a ? a ? 1 a ? 2 6 = 0 ! ( x ) = ( a ? 1  1 q x; a ? 2  1 q x ; q ) 1 ( a  1 1 x; a  1 2 x ; q ) 1 b a ( x  a 1 )( x  a 2 ) , b aa 1 a 2 6 = 0  a ? ( x  a ? 1 ) ,  a ? a ? 1 6 = 0 ! ( x ) = ( a ? 1  1 q x ; q ) 1 ( a  1 1 x; a  1 2 x ; q ) 1 _ a ? 6 = 0 ! ( x ) = 1 ( a  1 1 x; a  1 2 x ; q ) 1  a ( x  a 1 ) ,  aa 1 6 = 0 ! ( x ) = ( a ? 1  1 q x; a ? 2  1 q x ; q ) 1 ( a  1 1 x ; q ) 1 b a ( x  a 1 )( x  a 2 ) , b aa 1 a 2 6 = 0 _ a 6 = 0 ! ( x ) = ( a ? 1  1 q x; a ? 2  1 q x ; q ) 1 8 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus P o o : Sine  ( x ) = b a ( x  a 1 )( x  a 2 ) and  ? = q  1  + ( q  1  1) x = b a ? ( x  a ? 1 )( x  a ? 2 ), we ha e ( q H  ? )(0) = q  ? (0) =  (0), so he p olynomials q H  ? and  ha e he same indep enden e m. Using he p owe expansion o he p olynomials  and  ?  ( x ) = b ax 2 +  ax + _ a;  ? ( x ) = b a ? x 2 +  a ? x + _ a ? ; we ha e b a ? = q  1 b a + ( q  1  1) b b ,  a ? = q  1  a + ( q  1  1)  b and _ a ? = q  1 _ a , whe e, b b;  b a e he o eÆien o he p owe expansion o (see Eq. (1.7)). Thus, 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : deg  < 2 = ) b a = 0 = ) b a ? 6 = 0 = ) deg  ? = 2 ; deg  = 2 = ) b a 6 = 0 8 > > > > > > > > > < > > > > > > > > > : b b 6 =  b a 1  q = ) b a ? 6 = 0 = ) deg  ? = 2 ; b b =  b a 1  q = ) b a ? = 0 8 > > > < > > > :  b 6 =   a 1  q = ) deg  ? = 1 ;  b =   a 1  q = ) deg  ? = 1 : In all ases we an apply di e ly he ab o e lemma whih immedia ely leads us o he desi ed esul . No ie also ha all he ob ained un ions a e me omo phi and so, hey a e on inuous and non- anishing a x = 0, so we an supp ose wi hou any loss o gene ali y ha ! (0) = 1.  In he ase when  (0) = 0, i is easy o see ha  ? (0) = 0. This ase equi es a mo e de ail s udy. In he ollowing we should keep in mind ha o he quasi-deni eness o u  6 0 and  and should b e op ime p olynomials (see [18 ℄). P op osi ion 2.3 Le u be a q  lassial un ional sa is ying he dis ibu ional equa ion (1.5) wi h  = b ax 2 +  ax , j b a j + j  a j > 0 , and = b bx +  b , b b 6 = 0 . Then he ol lowing ases, ompa ible wi h he quasi-deni eness o u , appea : (a) I  = b ax 2 , b a 6 = 0 , hen, deg  ? = 2 and i s wo ze os a e die en , o deg  ? = 1 . (b) I  = b ax 2 +  ax , b a  a 6 = 0 , hen, deg  ? = 2 , o deg  ? = 1 . () I  =  ax ,  a 6 = 0 , hen, deg  ? = 2 . P o o : (a) Sine  = b ax 2 , hen = b bx +  b , wi h  b 6 = 0, o he wise di ides  . The e o e,  ? = ( q  1 b a + ( q  1  1) b b ) x 2 + ( q  1  1)  bx has a non- anishing o eÆien on x . I b b 6 =  b a 1  q hen b a ? 6 = 0 and deg  ? = 2 and  ? has wo die en ze os one o whih is lo a ed a he o igin. I b b =  b a 1  q hen deg  ? = 1. The o he wo ases a e p o en analogously.  The nex s ep is o nd he q  weigh un ions o all p ossible ases ao ding wi h he ab o e p op osi ion ( ememb e ha  (0) = 0 =  ? (0)). The e a e wo la ge lasses. Class I o esp onding o he ase when  and  ? ha e non- anishing e m on x and I I when hey ha e a anishing e m on x . I. We s a wi h he ase when  and  ? ha e no - anishing e m on x . In his ase he e a e h ee die en p ossibili ies (sub lasses): (a)  ( x ) = b ax ( x  a 1 ), b aa 1 6 = 0 and  ? ( x ) = b a ? x ( x  a ? 1 ), b a ? a ? 1 6 = 0, (b)  ( x ) = b ax ( x  a 1 ), b aa 1 6 = 0 and  ? ( x ) =  a ? x ,  a ? 6 = 0, ()  ( x ) =  ax ,  a 6 = 0 and  ? ( x ) = b a ? x ( x  a ? 1 ), b a ? a ? 1 6 = 0. R.  Al a ez-Noda se and J. C. Medem 9 To nd he o esp onding q  weigh un ions we will ew i e he quo ien q H  ? = = xq (H  ? ) 0 =x 0 , whe e  = x 0 and H  ? = x (H  ? ) 0 . In gene al,  0 (0) 6 = (H  ? ) 0 (0), so, in o de o apply a me ho d, simila o he one used o p o e Theo em 2.1, we will assume ha ! an b e ew i en on he o m ! = j x j  ! 0 ,  2 C n 0 g , whe e  is a ee pa ame e o b e ound. An s aigh o wa d alula ions show ha i ! sa ises a q  Pea son equa ion (2.6) hen ! 0 sa ises he equa ion  0 ! 0 = aq H ! 0 (H  ? ) 0 , whe e a = q  . So, ! 0 = H n ( ! 0 ) aq (H  ? ) 0  0 ; a = q  ; o  = Log q ( a ) ; whe e Log q deno es he p inipal loga i hm on he basis q , j q j < 1. In he ollowing, we will use he no a ion  a =  b aa 1 and  a ? =  b a ? a ? 1 . No ie ha , wi h his no a ion,  = b ax ( x  a 1 ) = b ax 2 +  ax and  ? = b a ? x ( x  a ? 1 ) = b a ? x 2 +  a ? x . (a) In his ase, aq (H  ? ) 0  0 = aq 2 b a ? a ? 1 ( a ? 1  1 q x  1) b aa 1 ( a  1 1 x  1) = aq 2  a ? (1  a ? 1  1 q x )  a (1  a  1 1 x ) : I we ho ose now, a suh ha aq (H  ? ) 0 (0) =  0 (0), i.e., aq 2  a ? =  a , o equi alen ly,  = Log q ( a ) =  2 + Log q  a  a ? , we an apply he Lemma 2.1 o ge , ! 0 = ! 0 (0) ( a ? 1  1 q x ; q ) 1 ( a  1 1 x ; q ) 1 , whih leads, wi hou any loss o gene ali y, o he ollowing weigh un ion (he e we supp ose ha ! 0 is on inuous and ! 0 (0) 6 = 0) ! ( x ) = j x j  ( a ? 1  1 q x ; q ) 1 ( a  1 1 x ; q ) 1 ;  = Log q ( a ) =  2 + Log q  a  a ? : (2.8) (b) In his ase, aq (H  ? ) 0  0 = aq 2  a ?  a (1  a  1 1 x ) = ) ! 0 = ! 0 (0) 1 ( a  1 1 x ; q ) 1 ;  = Log q ( a ) =  2 + Log q    a  a ?  ; so, ! ( x ) = j x j  ( a  1 1 x ; q ) 1 ;  = Log q ( a ) =  2 + Log q    a  a ?  : Finally, in he las ase (), we ob ain ! ( x ) = j x j  ( a ? 1  1 q x ; q ) 1 ;  = Log q ( a ) =  2 + Log q    a  a ?  : I I. Le onside he o he ase, i.e., when  and  ? ha e a anishing e m on x . In his ase he e a e wo p ossibili ies: (i) deg  6 = deg  ? whih is di ided in wo sub ases (a)  = b ax 2 ,  ? =  a ? x , and (b)  =  ax ,  ? = b a ? x 2 , and (ii) deg  = deg  ? , whih also is di ided in wo sub ases (a)  = b ax 2 ,  ? = b a ? x ( x  a ? 1 ), a ? 1 6 = 0, and (b)  = b ax ( x  a 1 ),  ? = b a ? x 2 , a 1 6 = 0. In b o h ases, he me ho d used in he ase I o non- anishing o eÆien s an no b e used. (i) In o de o sol e he p oblem o ase I I(i) we will gene alize an idea by Hake [12 ℄. Le us dene he un ion h (  ) : [0 ; 1 ) ! R dened by h (  ) ( x ) = p x log q x    ;  6 = 0 ; whih has he ollowing p op e y H h  = x  h  , o , equi alen ly, h  ( q x ) = x  h  ( x ), o all x  0. I we now dene he un ion ! = x  h (1) , hen, o he ase I I(i)a we ha e H ! = H x  h (1) = q  x  xh (1) = q  x! = ) x H ! = q  x 2 ! ; 16 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus The Ro d igues o mula is e y use ul o nding he explii exp ession o he p olynomials P n . In a , using he o mula  ? n ( x ) = q  n 2  + n (1  q ) n x n n X k =0 (  1) k q k ( k +1) 2  nk  n k  q ( q k  n x ) ;  n k  q = ( q ; q ) n ( q ; q ) k ( q ; q ) n  k ; whe e  n 2  = n ( n  1) 2 , one easily ob ains P n = q  n 2  n (1  q ) n x n n X k =0 (  1) k q k ( k +1) 2  nk  n k  q H ( n )  ( xq k  n ) ! ( q n  k x ) ! ( x ) ; o , equi alen ly, P n = n (  1) n (1  q ) n x n n X k =0 (  1) k q  k 2   n k  q H ( n )  ( xq  k ) ! ( q  k x ) ! ( x ) : Now, aking in o aoun he q  Pea son equa ion (2.6) H ! ! =  q H  ? () H  1 ! ! = q  ? H  1  ; we ob ain he ollowing explii exp ession o he q  lassial p olynomials in e ms o he p olyno- mials  and  ? : P n = n (  1) n (1  q ) n x n n X k =0 (  1) k q  k 2  + k  n k  q k  1 Y i =0  ? ( xq  i ) n  k  1 Y i =0  ( xq i ) : (3.6) This o mula is equi alen o he one ob ained in [5 , Eq. (4.14)℄, [23, Eq. (33)℄ and [2, Eq. (2.24)℄ o he q  p olynomials in he non-uni o m la ie x ( s ) =  1 q s . 3.3.2 The hyp e geome i ep esen a ion  We s a wi h he ; Jaobi/Jaobi amily, i.e., he ase when  = b a ( x  a 1 )( x  a 2 ) and  ? = b a ? ( x  a ? 1 )( x  a ? 2 ) b a ? a 1 a 2 b a ? a ? 1 a ? 2 6 = 0. The o he ases an b e ob ained in a simila way. Then, subs i u ing in he ab o e exp ession we nd ha he q  lassial p olynomials b eomes P n = n ( b aa 1 a 2 ) n ( x=a 1 ; q ) n ( x=a 2 ; q ) n (1  q ) n x n 3 ' 2 q  n ; a ? 1 x  1 ; a ? 2 x  1 q 1  n a 1 x  1 ; q 1  n a 2 x  1      q ; b a ? b a q  n +3 ! : F om he las o mula i is no easy o see ha P n a e p olynomials on x o deg ee exa ly equal n , hus, we will apply o he ab o e equa ion he ans o ma ions (3.2.5) and (3.2.3) gi en in [10, page 61℄. No ie ha we an apply he ans o ma ion o mula (3.2.5) [10, page 61℄ b eause he p olynomials  and q  ? ha e he same indep enden e m, and hen he ondi ion b aa 1 a 2 = q b a ? a ? 1 a ? 2 is ullled. So, he hyp e geome i ep esen a ion o he moni q  lassial ; Jaobi/Jaobi p olynomials is P n ( x ) = a n 2 ( a ? 1 =a 2 ; q ) n ( a ? 2 =a 2 ; q ) n ( a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; q ) n 3 ' 2 q  n ; a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; x=a 2 a ? 1 =a 2 ; a ? 2 =a 2      q ; q ! : (3.7) No ie ha , sine  and  ? a e in a ian wi h esp e o he hange a 1 () a 2 and a ? 1 () a ? 2 , hen we an ob ain an equi alen hyp e geome i ep esen a ion P n ( x ) = a n 2 ( a ? 1 =a 1 ; q ) n ( a ? 2 =a 1 ; q ) n ( a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; q ) n 3 ' 2 q  n ; a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; x=a 1 a ? 1 =a 1 ; a ? 2 =a 1      q ; q ! : (3.8) R.  Al a ez-Noda se and J. C. Medem 17 No ie also ha om any o he ab o e wo o mulas ollwos ha P n is a p olynomial o deg ee exa ly equal n . Be o e s a wi h he de ailed s udy o eah ase le us w i e ano he equi alen o m o he ; Jaobi/Jaobi p olynomials whih an b e ob ained applying he ans o ma ion (I I I.12) om [10, page 241-242℄ o (3.7): P n ( x ) = q  n 2  (  a ? 2 ) n ( a ? 1 =a 2 ; q ) n ( a ? 1 =a 1 ; q ) n ( a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; q ) n 3 ' 2 q  n ; a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; a ? 1 =x a ? 1 =a 2 ; a ? 1 =a 1      q ; q x=a ? 2 ! : (3.9) I we now ho ose  = aq ( x  1)( bx   ) and  ? = q  2 ( x  aq )( x  q ), hen Theo em 2.1 and Eq. (3.7) gi es, o he weigh un ion and he p olynomials, esp e i ely ! ( x ) = ( x=a; x= ; q ) 1 ( bx=; x ; q ) 1 ; p n ( x ; a; b;  ; q ) = ( aq ; q ) n ( q ; q ) n ( abq n +1 ; q ) n 3 ' 2 q  n ; abq n +1 ; x aq ; q      q ; q ! ; i.e., he Big q  Jaobi p olynomials. I we now ho ose  = q  N  1 hey b eomes he q  Hahn p olyno- mials Q n ( x ; a; b; N j q ) (usually hey a e w i en as p olynomials in x = q  s , see [13 , 18 ℄). Ob iously, i we use ins ead o o mula (3.7) he o mulas (3.8) and (3.9) we ob ain o he ep esen a ions o he Big q  Jaobi p olynomials. Fo he o he 11 ases we an do he same, subs i u e he p olynomials  and  ? in (3.6) and make he o esp onding alula ions, bu he e we will show how, om he q  lassial ; Jaobi/Jaobi p olynomials, an b e de i ed all o he ases by aking he app opia e limi s. A simila s udy ha e b een done in [23 ℄. He e we will omple e i . We will gi e he de ails only in some sp eial diÆul " ases o when he la i y and he au ay a e equi ed.  We on inue wi h he q  lassial ; Jaobi/Lague e p olynomials. To ob ain hem we ake he limi a ? 2 ! 1 . Then,  = b a ( x  a 1 )( x  a 2 ) and q  ? = q b a ? ( x  a ? 1 )( x  a ? 2 ) = q b a ? a ? 2 ( x  a ? 1 )( x=a ? 2  1) = b aa 1 a 2 a ? 1 ( x  a ? 1 )( x=a ? 2  1) !  b aa 1 a 2 a ? 1 ( x  a ? 1 ) ; whe e he ela ion b aa 1 a 2 = q b a ? a ? 1 a ? 2 has b een used. In his ase and sine lim a ? 2 !1 ( a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; q ) k ( a ? 2 =a 2 ; q ) k = q ( n  1) k  a ? 1 a 1  k ; Eq. (3.7) b eomes P n ( x ) =  a 1 a 2 a ? 1  n ( a ? 1 =a 2 ; q ) n q  n ( n  1) 2 ' 1 q  n ; x=a 2 a ? 1 =a 2      q ; q n a ? 1 =a 1 ! : (3.10) I we ho ose now  = ( x  1)( x + b ) and  ? = q  2  ( x  bq ), hen we ob ain he q  Meixne p olynomials M n ( x ; b;  ; q ) = (   ) n ( bq ; q ) n q  n 2 2 ' 1 q  n ; x bq      q ;  q n +1  ! : In his ase ! ( x ) = ( x=b ; q ) 1 (  x=b;x ; q ) 1 . Pu ing in he ab o e o mulas b = q  N  1 and  =  p  1 we a i e o he Quan um q  K a huk p olynomials K q m n ( x ; p; N ; q ).  The nex amily is he q  lassial ; Jaobi/He mi e one. In his ase we ake he limi a ? 1 ; a ? 2 ! 1 . Then,  = b a ( x  a 1 )( x  a 2 ) and q  ? = b a ? a 1 a 2 , hus (3.7) b eomes P n ( x ) = (  a 2 )  n q  n 2  2 ' 0 q  n ; x=a 2 |      q ; q n a 2 =a 1 ! : (3.11) 18 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus Cho osing  = ( x  a )( x  1) and q  ? = a we ob ain he Al-Salam & Ca li z I I p olynomials V ( a ) n ( x ; q ) = (  a ) n q   n 2  2 ' 0 q  n ; x 0      q ; q n a ! ; I now  = ( x  i )( x + i ) and q  ? = 1, we a i e o he Dis e e q  He mi e p olynomials I I e h n ( x ; q ) e h n ( x ; q ) = i  n 2 ' 0 q  n ; ix |      q ;  q  n ! = x n 2 ' 1 q  n ; q  n +1 0      q 2 ;  q 2 x 2 ! ; and o he weigh un ion we ha e ! ( x ) = ( ix;  ix ; q )  1 1 = (  x 2 ; q 2 ) 1 =  Q 1 k =0 (1 + x 2 q 2 k )   1 .  The q  lassial ; Lague e/Jaobi p olynomials. In his ase a 2 ! 1 . Then,  =  q b a ? a ? 1 a ? 2 a  1 1 ( x  a 1 ) and  ? = b a ? ( x  a ? 1 )( x  a ? 2 ), hus Eq. (3.9) gi es P n ( x ) = (  a ? 2 ) n q  n 2  ( a ? 1 =a 1 ; q ) n 2 ' 1 q  n ; a ? 1 =x a ? 1 =a 1      q ; q x=a ? 2 ! = a n 1 ( a ? 1 =a 1 ; q ) n ( a ? 2 =a 1 ; q ) n 3 ' 2 q  n ; x=a 1 ; 0 a ? 1 =a 1 ; a ? 2 =a 1      q ; q ! : (3.12) The las equali y ollows om he Jakson ans o ma ion o mula (see [10, Eq. (I I I.5), page 241℄), o , di e ly, aking he limi in o mula (3.8). I we now ho ose  =  aq ( x  1) and  ? = q  2 ( x  aq )( x  q ), we ob ain he Big q  Lague e p olynomials p n ( x ; a;  ; q ) = ( aq ; q ) n ( q ; q ) n 3 ' 2 q  n ; 0 ; x aq ; q      q ; q ! = ( aq ; q ) n (  q ) n q  n 2  2 ' 1 q  n ; aq x  1 aq      q ; x  ! : No ie ha hey a e no hing else ha he Big q  Jaobi when b = 0. He e ! ( x ) = ( x=a;x= ; q ) 1 ( x ; q ) 1 . To his lass also b elong he K a n ( x ; p; N ; q ). In a hey a e Big q  Lague e p olynomials wi h pa ame e s a = q  N  1 and  = p .  The q  lassial ; He mi e/Jaobi p olynomials. In his ase a 1 ; a 2 ! 1 , hus  = q b aa ? 1 a ? 2 and  ? = b a ? ( x  a ? 1 )( x  a ? 2 ). Then, om Eq. (3.9) one easily nd P n ( x ) = q  n 2  (  a ? 2 ) n 2 ' 1 q  n ; a ? 1 =x 0      q ; q x=a ? 2 ! : (3.13) Now ho osing  = a and q  ? = ( x  1)( x  a ), (3.13) leads o he Al-Salam & Ca li z I p olynomials U ( a ) n ( x ; q ) = (  a ) n q  n 2  2 ' 1 q  n ; x  1 0      q ; x q a ! : In his ase he q  weigh un ion akes he o m ! ( x ) = ( q x=a; q x ; q ) 1 . I we pu a =  1, he he Al-Salam & Ca li z I p olynomials b eomes he dis e e q  He mi e p olynomials I h n ( x ; q ). Fo he 0  amilies he si ua ion is mo e omplia e and a new pa ame e Æ should b e inluded. R.  Al a ez-Noda se and J. C. Medem 19  To ob ain he 0  Bessel/Jaobi p olynomials we will ake he limi a 1 ; a 2 ; a ? 2 ! 0. Thus,  = b ax 2 and  ? = b a ? ( x  a ? 1 ) x , bu now we ha e a p oblem aking he limi in he exp ession ( a ? 1 a ? 2 a  1 1 a  1 2 q n  1 ; q ) k , so we will obliged he pa ame e s a 1 ; a 2 ; a ? 2 end o ze o suh ha a ? 2 a  1 1 a  1 2 = q Æ , wi h Æ a xed ons an suh ha q Æ = b a= ( q b a ? a ? 1 ). Then, aking he limi in Eq. (3.7) we ob ain P n ( x ) = q  n 2  (  a ? 1 ) n ( q n + Æ  1 ; q ) n 2 ' 1 q  n ; q n + Æ  1 0      q ; q x=a ? 1 ! ; q Æ = b a q b a ? a ? 1 : (3.14) To his lass b elongs he Al e na i e q  Cha lie p olynomials K n ( x ; a; q ). In a , pu ing  = ax 2 and  ? = q  2 x (1  x ), hus q Æ =  aq and hen K n ( x ; a ; q ) = (  1) n q  n 2  (  aq n ; q ) n 2 ' 1 q  n ;  aq n 0      q ; q x ! : Fo hem we ha e ! ( x ) = j x j  ( x  1 ; q )  1 1 , whe e q  =  a=q .  Fo he 0  Bessel/Lague e p olynomials we ha e he limi a 1 ; a 2 ; a ? 1 ! 0 and a ? 2 ! 1 . Thus,  = b ax 2 and  ? = b a ? ( x  a ? 1 )( x  a ? 2 ) = b a ? a ? 2 ( x=a ? 2  1)( x  a ? 1 ) = b aa 1 a 2 a ? 1  1 q  1 ( x=a ? 2  1)( x  a ? 1 ). I we now ake he limi in suh a way ha a ? 1 a 1 a 2 =  q Æ we a i e o he un ion  ? = b aq  Æ  1 x . In his ase Eq. (3.9) immedia ely gi es P n ( x ) = q  n ( n + Æ  1) (  1) n 1 ' 1 q  n 0      q ;  q n + Æ x ! ; q Æ =  b a  a ? q : (3.15) Now, se ing  = x 2 and  ? = q  2 x , we ha e q Æ =  q and we ob ain he S iel jes-Wige p olyno- mials S n ( x ; q ) = (  1) n q  n 2 1 ' 1 q  n 0      q ;  xq n +1 ! : He e ! ( x ) = p x log q x  1 .  The 0  Jaobi/Jaobi p olynomials. In his ase he limi is a 2 ; a ? 2 ! 0 p o iding ha a ? 2 =a 2 = q Æ , hen  = b ax ( x  a 1 ),  ? = b a ? x ( x  a ? 1 ) and (3.7) gi es P n ( x ) = q  n 2  (  a ? 1 ) n ( q Æ ; q ) n ( a ? 1 =a 1 q Æ + n  1 ; q ) n 2 ' 1 q  n ; a ? 1 =a 1 q n + Æ  1 q Æ      q ; q x=a ? 1 ! ; q Æ = b aa 1 q b a ? a ? 1 : (3.16) Pu ing  = ax ( bq x  1) and  ? = q  2 x ( x  1), q Æ = aq , hus p n ( x ; a; b j q ) = (  1) n q  n 2  ( aq ; q ) n ( abq n +1 ; q ) n 2 ' 1 q  n ; abq n +1 aq      q ; q x ! ; whih a e no hing else ha he Li le q  Jaobi p olynomials. I now we ake  = px (1  x ),  ? = q  2 x ( x  q  N ) we a i e o he ollowing exp ession K n ( x ; p; N ; q ) = (  1) n q  nN +  n 2  (  pq N +1 ; q ) n (  pq n ; q ) n 2 ' 1 q  n ;  pq n  pq N +1      q ; xq N +1 ! ; ha s ons i u es an al e na i e deni ion o he q  K a huk p olynomials whih is equi alen o he mo e" s anda d one jus using he ans o ma ion o mula (I I I.7) om [10, page 241℄ K n ( x ; p; N ; q ) = ( q  N ; q ) n (  pq n ; q ) n 3 ' 2 q  n ; x;  pq n q  N ; 0      q ; q ! : 20 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus Finally, we ha e ! ( x ) = j x j  ( q x ; q ) 1 ( q bx ; q ) 1 , q  = a and ! ( x ) = j x j  ( q N +1 x ; q ) 1 ( x ; q ) 1 , q  = pq N o he weigh un ions o he Li le q  Jaobi and q  K a huk p olynomials, esp e i ely.  The 0  Jaobi/Lague e p olynomials. In his ase we ake he limi is a 2 ; a ? 2 ! 0 and a ? 1 ! 1 in suh a way ha a ? 2 =a 2 =  q Æ , so  = b ax ( x  a 1 ),  ? = b aa 1 q  Æ  1 x =  a ? x , and hen P n ( x ) = (  a 1 ) n q  n ( n + Æ  1) 2 ' 1 q  n ; x=a 1 0      q ;  q n + Æ ! ; q Æ = b aa 1 q  a ? : (3.17) Pu ing  = ax ( x + 1) and  ? = q  2 x , hen q Æ =  aq , and we ob ain he q  Lague e p olynomials L  n ( x ; q )  L n ( x ; a ; q ) L n ( x ; a ; q ) = (  1) n q  n 2 a  n 2 ' 1 q  n ;  x 0      q ; aq n +1 ! ; ! ( x ) = j x j  (  x ; q ) 1 ; q  =  a: I we now ho ose  = x ( x  1) and  ? = q  2 ax , we ob ain q Æ = q =a and hen we a i e o he q  Cha lie p olynomials C n ( x ; a ; q ) = (  1) n q  n 2 a n 2 ' 1 q  n ; x 0      q ;  q n +1 a ! ; ! ( x ) = j x j  ( x ; q ) 1 ; q  = a  1 :  The 0  Jaobi/Bessel p olynomials. He e we ake he limi is a 2 ; a ? 1 ; a ? 2 ! 0 in suh a way ha a ? 1 a ? 2 =a 2 = q Æ , so  = b ax ( x  a 1 ),  ? = b a ? x 2 = b aa 1 q  Æ  1 x 2 , and hen (3.7) gi es P n ( x ) = q n ( n + Æ  1) ( q n + Æ  1 =a 1 ; q )  1 n 2 ' 0 q  n ; q n + Æ  1 =a 1 |      q ; xq 1  Æ ! ; q Æ = b aa 1 q b a ? : (3.18) This amily do es no app ea in he q  Askey Sheme unless hey a e no a i ial limi o a mo e gene al q  amily. We will ake he ollowing pa ame e iza ion  = ax ( x  b ) and  ? = q  2 x 2 . Then, q Æ = abq and we ob ain ha his 0  Jaobi/Bessel p olynomials, deno ed by j n ( x ; a; b ) j n ( x ; a; b ) = ( ab ) n q n 2 ( aq n ; q )  1 n 2 ' 0 q  n ; aq n |      q ; x= ( ab ) ! ; ! ( x ) = j x j  ( bq =x ; q ) 1 ; q  = a  1 q  5 : They main da a a e shown in Table 3.3.2.  The 0  Lague e/Jaobi p olynomials. In his ase a 2 ; a ? 2 ! 0, a 1 ! 1 , q Æ =  a ? 2 =a 2 , hen  =  ax = b a ? a ? 1 q Æ +1 x ,  ? = b a ? x ( x  a ? 1 ), and P n ( x ) = (  a ? 1 ) n q  n 2  (  q Æ ; q ) n 2 ' 1 q  n ; 0  q Æ      q ; xq =a ? 1 ! ; q Æ =  a b a ? a ? 1 q : (3.19) Pu ing  =  ax and  ? = q  2 x ( x  1), hus q Æ =  aq , and we ob ain he Li le q  Lague e o Wall p olynomials p n ( x ; a j q ) = (  1) n q  n 2  ( aq ; q ) n 2 ' 1 q  n ; 0 aq      q ; q x ! ; ! ( x ) = j x j  ( q x ; q ) 1 ; q  =  a:  Finally, he 0  Lague e/Bessel amily ollows om Eq. (3.8) aking he limi a 1 ; a ? 1 ; a ? 2 ! 0 and a 2 ! 1 p o iding ha a ? 1 a ? 2 =a 1 =  q Æ , hus  =  ax = b a ? q Æ +1 x ,  ? = b a ? x 2 and P n ( x ) = (  1) n q n ( n + Æ  1) 2 ' 0 q  n ; 0 |      q ;  xq 1  Æ ! ; q Æ =  a q b a ? : (3.20) R.  Al a ez-Noda se and J. C. Medem 21 As he ase o 0  Jaobi/Bessel, his ase leads o a new amily whih is no in he q  Askey Tableau. In his ase we will adop he pa ame e iza ion  =  ax = b a ? ax ,  ? = q  2 x 2 , q Æ = aq , hus P n ( x )  l n ( x ; a ) = (  a ) n q n 2 2 ' 0 q  n ; 0 |      q ;  x=a ! ; ! ( x ) = j x j  p x log q x  1 +1 ; q  = a=q : Rema k 3.2 No ie ha in some examples he q  weigh un ions looks e y die en om he ones gi en in [13℄. Some imes he eason is he inde e mina eness o he assoia ed momen p ob- lem (e.g. he S iel jes-Wiege polynomials o he q  Lague e polynomials. Also, beause some imes ins ead he q  in eg als, dis e e sums a e used (see e.g. he example o he Li le q  Jaobi polyno- mials in [13℄). Table 3.3.2: The q  lassial p olynomials j n ( x ; a; b ) and l n ( x ; a ) P n j n ( x ; a; b ) l n ( x ; a )  ax ( x  b ) ax  ? q  2 x 2 q  2 x 2 abq +(1  aq ) x q (1  q ) aq  x ( q  1) q b  n  q  n [ n ℄( a + q n ) 1  q q  n [ n ℄ 1  q n q  n 2  + n (1  q ) n ( aq n ; q ) n q  n 2  + n (1  q ) n d n abq n ( 1  q n + aq 2 n  q n +1 ) (1  aq 2 n  1 )(1  aq 2 n +1 ) aq n  q n + q n +1  1  g n  a 2 b 2 q 3 n  1 (1  q n ) ( 1  aq n  1 ) (1  aq 2 n  1 ) 2 (1  aq 2 n )(1  aq 2 n  2 ) a 2 q 3 n  1 ( q n  1) a n a [ n ℄ 0 b n  ab [ n ℄(1  aq n ) ( 1+ aq 2 n ) (1  aq 2 n  1 )(1  aq 2 n +1 ) a [ n ℄  n a 2 b 2 q 2 n  1 [ n ℄(1  aq n ) ( 1  aq n  1 ) (1  aq 2 n  1 ) 2 (1  aq 2 n )(1  aq 2 n  2 ) a 2 q 2 n  1 [ n ℄ e n abq n (1  q n ) ( 1+ aq 2 n ) (1  aq 2 n  1 )(1  aq 1+2 n ) aq n ( q n  1) h n a 3 b 2 q 4 n  2 (1  q n ) ( 1  q n  1 ) (1  aq 2 n  1 ) 2 (1  aq 2 n )(1  aq 2 n  2 ) 0 d 0 n abq n +1 ( 1  q n  q n +1 + aq 2 n +2 ) (1  aq 2 n +1 )(1  aq 2 n +3 ) aq n +1  q n + q n +1  1  g 0 n  a 2 b 2 q 3 n +1 (1  q n ) ( 1  aq n +1 ) (1  aq 2 n )(1  aq 2 n +1 ) 2 (1  aq 2 n +2 ) a 2 q 3 n +1 ( q n  1) No ie ha o all 0 < q < 1, he p olynomials l n ( x ; a ) ne e ons i u es a p osi i e deni e amily sine g n < 0 (see he Fa a d heo em (1.1)). The ase i he j n ( x ; a; b ) p olynomials is mo e omplia ed. Ne e heless, ho osing a = q  N i is easy o show ha j n ( x ; a; b ) ons i u e a ni e amily (simila o he q  Hahn p olynomials) whih is p osi i e deni e sine g n > 0 o all n = 0 ; 1 ;::: ; [ N = 2℄. The de ailed s udy o he p osi i e deni e ases in dep endene o he o o s o 22 q  Classial polynomials and he q  Askey and Niki o o -U a o Tableaus  and  ? will b e onside ed in a o homing pap e . Aknowledgemen s: This wo k has b een pa ially supp o ed by he Jun a de Andalua (FQM- 207), he Eu op ean p o je INTAS-93-219-ex and by he Spanish Di eion Gene al de Ense ~nanza Sup e io (DGES) g an s PB-96-0120-C01-01. We hanks J. S. Dehesa and F. Ma ellan o help ull diussions and ema ks. Re e enes [1℄ L. Ahl o s, Complex Analysis. MG aw-Hill, New Yo k, 1953. [2℄ R.  Al a ez-Noda se and J. A es u, On he q- p olynomials in he exp onen ial la ie x ( s ) =  1 q s +  3 . In eg al T ans o m. 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