Mellin transforms for some families of q-polynomials

MELLIN TRANSFORMS FOR SOME FAMILIES OF q -POLYNOMIALS Renato  Alvarez-No darse a M. K. Atakishiyeva b N. M. Atakishiyev  a Departamento de Analisis Matematio, Universidad. de Sevil la, Apdo. 1160, E-41080 Sevil la, and Instituto Carlos I de Fsia Teoria y Computaional, Universidad de Granada, E-18071, Granada, Spain. E-mail: ranus.es b Faultad de Cienias, UAEM, Apartado Postal 396-3, CP 62250, Cuernavaa, Morelos, Mexio. E-mail: mesumaservm.f.uaem.mx  Instituto de Matematias, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaa, Morelos, Mexio. E-mail: natigmatuer.unam.mx Abstrat By using Ramanujan's q -extension of the Euler integral representation for the gamma funtion, we derive the Mellin integral transforms for the families of the dis- rete q -Hermite I I, the Al-Salam{Carlitz I I, the big q -Laguerre, the big q -Legendre, the big q -Jaobi and the q -Hahn p olynomials. Key words: Mellin integral transforms, q -p olynomials. 1 Intro dution Mellin integral transforms for some families of basi hyp ergeometri p olynomials from the Askey sheme [15℄ were onsidered in [7℄. Derivation of 1 This researh has b een supp orted in part by the Ministerio de Cienias y Te- nologa of Spain under the grant BFM-2000-0206-C04-02, the Junta de Andalua under grant FQM-262, the Europ ean pro jet INTAS 2000-272, and by the Mexian UNAM-DGAPA pro jet IN112300. Two of us (MKA and NMA) are most grateful to the Faultad de Matematias, Universidad de Sevilla for the hospitality extended to them during their visit to Sevilla in May-June 2001, when this work was ompleted. Preprint submitted to Elsevier Preprint 13 Otob er 2003 these Mellin transform pairs is essentially based on the use of Ramanujan's q - extension [17,4,5℄ of the Euler integral representation for the gamma funtion ( x ) (1  x )  q (1  x ) = 1 Z 0 t x  1 dt E q ((1  q ) t ) ; < x > 0 ; (1) where  q ( z ) is the q -gamma funtion  q ( x ) := ( q ; q ) 1 ( q x ; q ) 1 (1  q ) 1  x ; 0 < q < 1 ; and E q ( z ) is Jakson's q -exp onential funtion E q ( z ) := 1 X n =0 q ( n 2 ) ( q ; q ) n z n = (  z ; q ) 1 ; (2) where  n 2  = n ( n  1) = 2. We employ the standard notation of the q -sp eial funtions theory, see e.g. [12℄ or [2℄. In partiular, the q -shifted fatorials are given by ( a ; q ) 0 = 1 ; ( a ; q ) n = n  1 Y k =0 (1  aq k ) ; n = 1 ; 2 ;:::; ( a ; q ) 1 = 1 Y k =0 (1  aq k ) ; (3) and we will use the notation r  p 0 B  a 1 ; : : : ; a r b 1 ; : : : ; b p      q ; z 1 C A := 1 X k =0 ( a 1 ; q ) k  ( a r ; q ) k ( b 1 ; q ) k  ( b p ; q ) k z k ( q ; q ) k  (  1) k q ( n 2 )  p  r +1 (4) for the basi hyp ergeometri series. It is well known that Ramanujan evaluated a numb er of integrals that extend the lassial b eta integral of Euler (see [17℄, [13℄, [4℄-[6℄). These integrals have asso iated orthogonal p olynomials that have played a signiant role in the development of the q -sp eial funtions theory. Ramanujan's now lassial q - extension of the b eta integral of Euler is 1 Z 0 x  E q ( q b +  x ) E q ( x ) dx = (  ) (1   )  q ( b )  q (1   )  q ( b +  ) : (5) 2 The formula (1) is an easy onsequene of (5) and the limit relation lim b !1  q ( b )  q ( b +  ) = (1  q )  ; when the hange of variables x = (1  q ) t is made. It was shown in [7℄ that by using a q -analogue of Euler's reetion formula  q ( x )  q (1  x ) = iq 1 = 8 (1  q ) ( q ; q ) 3 1 q x= 2  1 (ln q  ix= 2 ; q 1 = 2 ) ;  1 ( z ; q ) is the theta-funtion of Jaobi, one an represent (1) in the form  q ( x )  q ( x ) = 1  q ln q  1 q x ( x  1) = 2 1 Z 0 t x  1 dt E q ((1  q ) t ) ; < x > 0 ; (6) where  q ( x ) is some p erio di fator, i.e.  q ( x + n ) =  q ( x ) for any non-negative integer n (an expliit form of  q ( x ) an b e found in [7℄). It should b e emphasized that this formula is simply Jaobi's triple pro dut identity for the theta-funtion  1 ( z ; q ), rewritten in terms of the q -gamma funtion  q ( z ). The observation that Jaobi's triple pro dut identity is a q -analogue of Euler's reetion formula for the gamma funtion ( z ) was known to G. Andrews and R. Askey sine \the 1975-1976 aademi year". Unfortunately, they never published anything ab out this fat. Besides, R. Askey b elieves that this observation \is due to George Andrews" (see the very end of [3℄), although G. Andrews omments that \Dik mo destly attributes it to me; this is more a measure of his generosity than his aurate memory" (e-mail ommuniation, Novemb er 16, 2001). Anyway, the idea of regarding Jaobi's triple pro dut identity as a q -extension of Euler's reetion formula is at least 25 years old. Two of us (MKA and NMA) regret that we were not already aware of this fat at the time of the writing of [7℄ (whih was the starting p oint for using this idea in the derivation of Mellin integral transforms for some families of q -p olynomials). In view of the p erio diity of  q ( x ), one an then derive from (6) a Mellin integral transform for the pro dut p n ( t ; q ) E  1 q ((1  q ) t ), where p n ( z ; q ) is some p olynomial in z of degree n . In this way the Mellin integral transforms were obtained for all those families of q -p olynomials from the Askey sheme, in whih indep endent variable is the argument of an appropriate terminating basi hyp ergeometri series. They onsist of the Stieltjes{Wigert, the Rogers{ Szego, the q -Laguerre, the Wall, the alternative q -Charlier, and the little q - Jaobi p olynomials. In this pap er we wish to apply the tehnique of [7℄ to the study of those families of q -p olynomials from the Askey sheme, whih ontain the indep endent 3 variable x in one of the parameters of the orresp onding basi hyp ergeometri series. The simplest example of this typ e is the disrete q -Hermite I I p olynomials ~ h n ( x ; q ) := i  n q  ( n 2 ) 2  0 0 B  q  n ; ix |      q ;  q n 1 C A = i  n q  ( n 2 ) n X k =0 ( q  n ; q ) k ( ix ; q ) k ( q ; q ) k q nk  ( k 2 ) ; there are also the Al-Salam{Carlitz I I, the big q -Laguerre, the big q -Legendre, the big q -Jaobi, and the q -Hahn p olynomials (see [15℄). Our motivation for the study of q -analogues of Mellin integral transforms omes from mathematial physis. It is well known that in nonrelativisti quantum mehanis the o ordinate and momentum realizations are interrelated by the Fourier integral transform. But in some relativisti approahes to quantum mehanis the passage from the momentum to the onguration realization is aomplished by the Mellin integral transform. For instane, a relativisti quasip otential [14,16℄ mo del of the linear harmoni osillator, studied in detail in [8,9,11℄, is governed by a dierene Hamiltonian in the onguration x -realization. The passage from the onguration to the momentum realization is equivalent to the Mellin integral transform in the light-front variable p + = p 0 + p , rather than the Fourier transform as in the nonrelativisti ase. Therefore we b elieve that our tehnique an b e applied to onstruting various q -extensions of suh quantum-mehanial mo dels, based on dierene equations. 2 The Mellin transform for a partiular family of q -p olynomials It is well known that Euler's integral representation ( x ) = 1 Z 0 t x  1 e  t dt ; < x > 0 ; (7) for the gamma funtion ( x ) is an instane of the Mellin integral transform g ( z ) = M f f ( t ); z g = 1 Z 0 f ( t ) t z  1 dt 4 of the exp onential funtion f ( t ) = e  t . Similarly, (1) or (6) is the Mellin transform of the funtion E  1 q ((1  q ) t ): g q ( x ) := M f E  1 q ((1  q ) t ); x g = ln q  1 1  q  q ( x ) q  x ( x  1) = 2  q ( x ) ; < x > 0 : Let p n ( x ; q ) = n X k =0 a nk ( q ) ( x ; q ) k (8) b e a q -p olynomial of degree n in x (with o eÆients a nk ( q ), whih may dep end, in addition to q , on some other parameters ). Then M f p n ( t ; q ) E  1 q ((1  q ) t ); x g = n X k =0 a nk ( q ) M f ( t ; q ) k E  1 q ((1  q ) t ); x g ; where  is a onstant. Sine the q -shifted fatorial ( z ; q ) k = k X j =0 " k j # q q ( j 2 ) (  z ) j ; " k j # q := ( q ; q ) k ( q ; q ) k  j ( q ; q ) j ; is a p olynomial of degree k in z , we have M f p n ( t ; q ) E  1 q ((1  q ) t ); x g = n X k =0 a nk ( q ) k X j =0 " k j # q q ( j 2 ) (   ) j M f t j E  1 q ((1  q ) t ); x g = n X k =0 a nk ( q ) k X j =0 " k j # q q ( j 2 ) (   ) j g q ( x + j ) : (9) But g q ( x + j ) = ln q  1 1  q  q ( x ) q  ( x + j )( x + j  1) = 2  q ( x + j ) = q  j x  ( j 2 ) ( q x ; q ) j (1  q ) j g q ( x ) ; (10) b eause  q ( x + j ) = ( q x ; q ) j (1  q ) j  q ( x ) 5 by the denition of the q -gamma funtion  q ( x ). Substituting (10) into the right-hand side of (9), one thus obtains that M f p n ( t ; q ) E  1 q ((1  q ) t ); x g = g q ( x ) n X k =0 a nk ( q ) k X j =0 " k j # q ( q x ; q ) j  q  x 1  q ! j : (11) Finally, the q -binomial o eÆient h k j i q an b e written as " k j # q = (  1) j q k j  ( j 2 ) ( q  k ; q ) j ( q ; q ) j : (12) Therefore the sum over the index j in the right-hand side of (11) represents a terminating basi hyp ergeometri series 2  0 and we obtain the desired result M f p n ( t ; q ) E  1 q ((1  q ) t ); x g = g q ( x ) n X k =0 a nk ( q ) 2  0 0 B  q  k ; q x |      q ;  q k  x 1  q 1 C A : (13) This formula gives an expliit form of the Mellin integral transform for the funtion E  1 q ((1  q ) t ), multiplied by a p olynomial p n ( t ; q ) of the typ e (8) (with an arbitrary onstant  ). Observe that the right-hand side of (13) is a p olynomial of degree n in the variable q  x , times the funtion g q ( x ). Also, the terminating basi hyp ergeometri series 2  0 in (13) an b e written as 2  0 0 B  q  k ; q x |      q ;  q k  x 1  q 1 C A = C k q  x ; 1  q  q k ; q  1 ! ; where the q  1 -Charlier p olynomials C n ( z ; a ; q  1 ) are dened (f. [15℄, p. 112) as C n ( z ; a ; q  1 ) := 2  0 0 B  q  n ; z  1 |      q ;  z =a 1 C A : (14) For a partiular hoie of the onstant  , the sum over j in (11) is simplied; in other words, the terminating basi hyp ergeometri series 2  0 in (13) redues 6 to a monomial in the variable q  x . The p oint is that 2  0 0 B  q  n ; z  1 |      q ; z q n 1 C A = z n : (15) To verify (15), simply reverse the order of summation in the denition of 2  0 and use the limit ase of the q -Chu{Vandermonde sum 2  1 0 B  q  n ; b       q ; q 1 C A = ( =b ; q ) n (  ; q ) n b n with the vanishing parameter b . We note that the relation (15) an b e expressed in terms of the q  1 -Charlier p olynomials (14) as C n ( z ;  q  n ; q  1 ) = z n : From (15) it follows that if one ho oses  = q  1, then the Mellin integral transform (13) redues to M f p n (( q  1) t ; q ) E  1 q ((1  q ) t ); x g = g q ( x ) n X k =0 a nk ( q ) q  k x : (16) Notie that suh simpliation of (13) in the ase when  = q  1 is a onsequene of the following prop erty E q ( z ) = (  z ; q ) k E q ( q k z ) (17) of Jakson's q -exp onential funtion (2). Indeed, by the denition (8), M f p n (( q  1) t ; q ) E  1 q ((1  q ) t ); x g = n X k =0 a nk ( q ) 1 Z 0 (( q  1) t ; q ) k E q ((1  q ) t ) t x  1 dt = n X k =0 a nk ( q ) 1 Z 0 t x  1 dt E q ((1  q ) q k t ) ; (18) where we have employed the prop erty (17) with z = (1  q ) t . The hange of the variable t ! q  k t in (18) leads immediately to the Mellin transform (16). Now it remains only to onsider onrete examples of the aforementioned families of q -p olynomials from the Askey sheme. 7 3 Conrete examples 1. We start with the Al-Salam{Carlitz I I p olynomials from the q -Askey sheme (see [15℄, p. 114) V ( a ) n ( x ; q ) := (  a ) n q  ( n 2 ) 2  0 0 B  q  n ; x |      q ; q n a 1 C A = (  a ) n q  ( n 2 ) n X k =0 ( q  n ; q ) k ( x ; q ) k ( q ; q ) k q k [ n  ( k  1) = 2℄ (  a  1 ) k ; (19) whih o upy the seond (i.e. next-to-lowest) level in the Askey sheme of basi hyp ergeometri p olynomials with the disrete orthogonality prop erty (see [15℄, p. 62). From (19) it follows that the o eÆients a nk ( q ) in (8) in this partiular ase are equal to a nk ( q ) = (  a ) n q  ( n 2 ) a n  k  n k  q ; (20) where we have used the relation (12). Substituting (20) into (16), we thus obtain a Mellin integral transform M f V ( a ) n (( q  1) t ; q ) E  1 q ((1  q ) t ); x g = (  a ) n q  ( n 2 ) H n ( q  x =a ; q ) g q ( x ) ; (21) where the Rogers{Szego p olynomials H n ( z ; q ) are the q -analogue of Hermite p olynomials on the unit irle (see [18,1,10℄), dened as H n ( z ; q ) := n X k =0  n k  q z k = 2  0 0 B  q  n ; 0 |      q ; z q n 1 C A : (22) Notie that the sp eial ase of the Al-Salam{Carlitz I I p olynomials (19) with a =  1 is known as the disrete q -Hermite I I p olynomials e h n ( x ; q ) (see [15℄, p. 119). Therefore, from (21) one obtains a Mellin transform M f e h n ( i (1  q ) x ; q ) E  1 q ((1  q ) t ); x g = i  n q  ( n 2 ) H n (  q  x ; q ) g q ( x ) ; whih interrelates the disrete q -Hermite I I and the Rogers{Szego p olynomials. From (22) it is evident that lim q ! 1  H n ( z ; q ) = ( z + 1) n : 8 The Mellin transform (21) in the limit as the parameter q ! 1  thus oinides with Euler's integral representation for the gamma funtion (7), b oth sides of whih are multiplied by the onstant fator (  1) n (1 + a ) n . 2. At the third level of the q -Askey sheme with the disrete orthogonality there is only one family of q -p olynomials of the typ e (8), namely the big q - Laguerre p olynomials (see [15℄, p. 91) P n ( x ; a; b ; q ) := 3  2 0 B  q  n ; 0 ; x aq ; bq      q ; q 1 C A = n X k =0 ( q  n ; q ) k ( x ; q ) k q k ( aq ; q ) k ( bq ; q ) k ( q ; q ) k : (23) The o eÆients a nk ( q ) in this ase are equal to a nk ( q ) = ( q  n ; q ) k q k ( aq ; q ) k ( bq ; q ) k ( q ; q ) k : (24) Substituting (24) into (16), one obtains the following Mellin integral transform M f P n (( q  1) t ; a; b ; q ) E  1 q ((1  q ) t ); x g = 3  2 0 B  q  n ; 0 ; 0 aq ; bq      q ; q 1  x 1 C A g q ( x ) (25) for the big q -Laguerre p olynomials P n ( x ; a; b ; q ). From the denition (23) it is lear that lim q ! 1  P n (( q  1) t ; a; b ; q ) = [1  (1  a )  1 (1  b )  1 ℄ n : Therefore the Mellin transform (25) in the limit as q ! 1  oinides with (7), multiplied by the onstant fator [1  (1  a )  1 (1  b )  1 ℄ n . 3. The big q -Jaobi p olynomials (see [15℄, p. 73) P n ( x ; a; b;  ; q ) := 3  2 0 B  q  n ; abq n +1 ; x aq ; q      q ; q 1 C A (26) o upy the fourth level in the Askey sheme for q -p olynomials with the disrete orthogonality. Taking into aount that the o eÆients a nk ( q ) in this ase are equal to a nk ( q ) = ( q  n ; q ) k ( abq n +1 ; q ) k q k ( aq ; q ) k ( q ; q ) k ( q ; q ) k ; 9