On a q-extension of the Hermite polynomials Hn(x) with the continuous orthogonality property on R1
Abstract
We study a polynomial sequence of q-extensions of the classical Hermite polynomials Hn(x), which satisfi es continuous orthogonality on the whole real line R with respect to the positive weight function. This sequence can be expressed either in terms of the q-Laguerre polynomials L n (x; q), = 1=2, or through the discrete q-Hermite polynomials ~hn(x; q) of type II.
Full text
On a q -extension of the Hermite p olynomials H n ( x ) with the ontinuous orthogonality prop erty on R 1 R. Alvarez-No darse , M. K. Atakishiyeva y , and N. M. Atakishiyev z Departamento de Analisis Matematio, Universidad de Sevilla, Ap do. 1160, E-41080 Sevilla, Spain and Instituto Carlos I de Fsia Teoria y Computaional, Universidad de Granada, E-18071 Granada, Spain E-mail: ranus.es y Faultad de Cienias, UAEM, Apartado Postal 396-3, CP 62250, Cuernavaa, Morelos, Mexio E-mail: mesumaservm.f.uaem.mx z Instituto de Matematias, UNAM, Apartado Postal 273-3, C.P. 62210 Cuernavaa, Morelos, Mexio E-mail: natigmatuer.unam.mx Abstrat We study a p olynomial sequene of q -extensions of the lassial Hermite p olynomials H n ( x ), whih satises ontinuous orthogonality on the whole real line R with respet to the p ositive weight funtion. This sequene an be expressed either in terms of the q -Laguerre p olynomials L ( ) n ( x ; q ), = 1 = 2, or through the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I. 1 Intro dution There is a well-known family of the ontinuous q -Hermite p olynomials of Rogers [1, 2℄ H n (os j q ) := n X k =0 n k q e i ( n 2 k ) = e in 2 0 q n ; 0 | q ; q n e 2 i ! ; (1.1) whih are q -extensions of the lassial Hermite p olynomials H n ( x ) for q 2 (0 ; 1). Throughout this pap er we will employ the standard notations of the q -sp eial funtions theory , see [3 ℄-[5 ℄. In partiular, n k q := ( q ; q ) n ( q ; q ) k ( q ; q ) n k = n n k q (1.2) stands for the q -binomial o eÆient and ( a ; q ) 0 = 1 and ( a ; q ) n = Q n 1 j =0 (1 aq j ) ; n = 1 ; 2 ; 3 ;::: , is the q -shifted fatorial. Besides, expliit forms of q -p olynomials from the Askeysheme [4℄ are often expressed in terms of the terminating basi hyp ergeometri p olynomial r s q n ; a 2 ;::: ;a r b 1 ; b 2 ;::: ;b s q ; z := n X k =0 ( q n ; q ) k ( a 2 ; q ) k ( a r ; q ) k ( b 1 ; q ) k ( b 2 ; q ) k ( b s ; q ) k z k ( q ; q ) k h ( 1) k q k ( k 1) = 2 i s r +1 (1.3) 1 Boletn de la So iedad Mexiana de Matematias 8 (2002), 221{232.
2 R. Alvarez-No darse et. al. On a q -extension of the Hermite polynomials H n ( x ) of degree n in the variable z . So the ontinuous q -Hermite p olynomials in (1) orresp ond to the ase when r = 2 and s = 0. The ontinuous q -Hermite p olynomials H n ( x j q ) satisfy the three-term reurrene relation 2 xH n ( x j q ) = H n +1 ( x j q ) + (1 q n ) H n 1 ( x j q ) : (1.4) They are orthogonal on the nite interval x 2 [ 1 ; 1℄ with resp et to the ontinuous weight funtion [2 ℄ w ( x ) = 1 p 1 x 2 1 Y k =0 h 1 + 2(1 2 x 2 ) q k + q 2 k i : (1.5) In the limit as q ! 1, they oinide with the ordinary Hermite p olynomials H n ( x ), i.e. , lim q ! 1 1 q 2 n= 2 H n r 1 q 2 x q ! = H n ( x ) : (1.6) There is another family, alled the disrete q -Hermite p olynomials of typ e I I , whih is a q -extension of the sequene of the Hermite p olynomials H n ( x ) [6 ℄. Their expliit form is given (see [4, p. 119℄) by ~ h n ( x ; q ) := i n q n ( n 1) = 2 2 0 q n ; i x | q ; q n ! = x n 2 1 q n ; q 1 n 0 q 2 ; q 2 x 2 ! : (1.7) The disrete q -Hermite p olynomials of typ e I I, ~ h n ( x ; q ), satisfy the three-term reurrene relation x ~ h n ( x ; q ) = ~ h n +1 ( x ; q ) + q 1 2 n (1 q n ) ~ h n 1 ( x ; q ) ; (1.8) but, ontrary to the p olynomials H n ( x j q ) of Rogers (1.1), the sequene f ~ h n ( x ; q ) g is orthogonal on the innite interval x 2 R with resp et to the disrete weight funtion, supp orted on the p oints x = q k ; > 0 ; k 2 Z . In the limit as q ! 1, the ~ h n ( x; q ) redue as well to the Hermite p olynomials H n ( x ): lim q ! 1 (1 q 2 ) n= 2 ~ h n ( x p 1 q 2 ; q ) = 2 n H n ( x ) : (1.9) Now we are in a p osition to formulate the aim of this pap er. We wish to nd suh q - extensions of the Hermite p olynomials H n ( x ), whih satisfy the following requirements: 1. They are p olynomials in the variable x , whih ob ey a three-term reurrene relation; 2. They are orthogonal on the whole real line R with resp et to a ontinuous p ositive weight funtion; 3. In the limit as q ! 1 they oinide with the Hermite p olynomials H n ( x ). Suh a family is of great interest from the p oint of view of p ossible appliations in mathematial physis. We remind the reader that the two most fundamental problems in nonrelativisti quantum mehanis, the harmoni osillator and the Coulomb system, are dened on R 3 [7℄. Thus our goal is equivalent to having suh a q -deformed version of the linear harmoni osillator, whih is still dened on the whole real line R and enjoys the ontinuous orthogonality prop erty on R with resp et to a p ositive weight funtion. 2 We will not pursue this viewp oint here. Instead we now fo us on the mathematial prop erties of the q -extensions of the Hermite p olynomials H n ( x ) under disussion. 2 It should b e noted that there are, in fat, several publiations (see [8 ℄{[16℄ and referenes therein) devoted to the study of expliit realizations, whih represent q -extensions of the Hermite funtions (or the wave funtions of the linear harmoni osillator) H n ( x ) e x 2 = 2 . But none of these realizations satises all of the aforementioned requirements : the ontinuous weight funtions in [8, 10, 13℄ are supp orted on the nite intervals; the ontinuous weight funtions in [9 , 15℄ are not p ositive; the q -extensions in [9℄, [12 ℄{[15 ℄ are not expressed in terms of p olynomials in the indep endent variable; and, nally, the orthogonality relations in [11℄{[13 ℄, [16℄ are disrete.
Boletn de la So iedad Mexiana de Matematias 8 (2002), 221{232 3 2 Denition of the sequene fH n ( x ; q ) g It is known that the Hermite p olynomials H n ( x ) an b e expressed through the Laguerre p olynomials L ( ) n ( x ) as H 2 n ( x ) = ( 1) n 2 2 n n ! L ( 1 = 2) n ( x 2 ) ; H 2 n +1 ( x ) = ( 1) n 2 2 n +1 n ! x L (1 = 2) n ( x 2 ) ; (2.1) where L ( ) n ( z ) := ( + 1) n n ! 1 F 1 n + 1 z = ( + 1) n n ! n X k =0 ( n ) k ( + 1) k z k k ! ; (2.2) and ( a ) n = ( a + n ) = ( a ) ; n = 0 ; 1 ; 2 ;::: , is the shifted fatorial. It is also known that a q -extension of the Laguerre p olynomials 3 L ( ) n ( x ; q ), dened [17℄-[19 ℄ as L ( ) n ( x ; q ) = ( q +1 ; q ) n ( q ; q ) n 1 1 q n q +1 q ; q n + +1 x ! = 1 ( q ; q ) n 2 1 q n ; x 0 q ; q n + +1 ! ; (2.3) satises two kinds of orthogonality relations, an absolutely ontinuous one and a disrete one. The former orthogonality relation 4 is given by 1 Z 0 x E q ( x ) L ( ) m ( x ; q ) L ( ) n ( x ; q ) dx = d 1 n ( ) Æ mn ; > 1 ; (2.4) where E q ( x ) is the Jakson q -exp onential funtion, E q ( z ) := 1 X n =0 q n ( n 1) = 2 ( q ; q ) n z n = ( z ; q ) 1 ; (2.5) and the normalization onstant d n ( ) is equal to d n ( ) = q n ( q ; q ) n ( q +1 ; q ) n ( q ; q ) 1 ( q ; q ) 1 sin ( + 1) : (2.6) It remains only to remind the reader, that in the limit as q ! 1 we have lim q ! 1 L ( ) n ((1 q ) x ; q ) = L ( ) n ( x ) : (2.7) 3 There are also the ontinuous q -Laguerre polynomials P ( ) n ( x j q ) with the orthogonality on the nite interval x 2 [ 1 ; 1℄ and the Wall ( or the little q -Laguerre ) p olynomials p n ( x ; a j q ) with the disrete orthogonality on the p oints x = q k ; k = 0 ; 1 ; 2 ;::: (see [4℄, pp. 105 and 107, resp etively). 4 It is worth noting that the fat of the integrability of the weight funtion x =E q ( x ) in (2.4) diretly follows from the integral 1 Z 0 t x 1 dt E q ((1 q ) t ) = ( x ) (1 x ) q (1 x ) ; Re x > 0 ; whih is a partiular ase of the Ramanujan integral extension of the beta funtion [20 ℄.
4 R. Alvarez-No darse et. al. On a q -extension of the Hermite polynomials H n ( x ) We an now dene, in omplete analogy with the relationship (2.1), a family of q - extensions of the Hermite p olynomials of the form H 2 n ( x ; q ) := ( 1) n ( q ; q ) n L ( 1 = 2) n ( x 2 ; q ) ; H 2 n +1 ( x ; q ) := ( 1) n ( q ; q ) n x L (1 = 2) n ( x 2 ; q ) : (2.8) With the aid of the limit relation lim q ! 1 ( q a ; q ) n (1 q ) n = ( a ) n (2.9) it is not diÆult to verify that from (2.8) and (2.3) one obtains lim q ! 1 (1 q ) n= 2 H n ( p 1 q x ; q ) = 2 n H n ( x ) ; (2.10) i.e. , the p olynomials H n ( x ; q ) so dened are indeed q -extensions of the Hermite p olynomials H n ( x ). The next step is to establish a ontinuous orthogonality relation and a three-term reurrene relation for the q -p olynomials H n ( x ; q ). 3 Orthogonality relation We b egin this setion with the following theorem: Theorem 1 The sequene of the q -polynomials fH n ( x ; q ) g , whih are dened by the relations (2.8), satises the orthogonality relation 1 Z 1 H m ( x ; q ) H n ( x ; q ) dx E q ( x 2 ) = q n 2 ( q 1 = 2 ; q 1 = 2 ) n ( q 1 = 2 ; q ) 1 = 2 Æ mn (3.1) on the whole real line R with respet to the ontinuous positive weight funtion w ( x ) = 1 =E q ( x 2 ) . Pro of . Sine the weight funtion in (3.1) is an even funtion of the indep endent variable x and H n ( x ; q ) = ( 1) n H n ( x ; q ) by the denition (2.8), the q -p olynomials of an even degree H 2 m ( x ; q ) and of an o dd degree H 2 n +1 ( x ; q ) ; m; n = 0 ; 1 ; 2 ;::: , are evidently orthogonal to eah other. Consequently, it suÆes to prove only those ases in (3.1), when degrees of p olynomials m and n are either simultaneously even or o dd. Let us onsider rst the former ase. From (2.8) and (2.4) it follows that 1 Z 1 H 2 m ( x ; q ) H 2 n ( x ; q ) dx E q ( x 2 ) = 2( 1) m + n ( q ; q ) m ( q ; q ) n 1 Z 0 L ( 1 = 2) m ( x 2 ; q ) L ( 1 = 2) n ( x 2 ; q ) dx E q ( x 2 ) = 2( 1) m + n ( q ; q ) m ( q ; q ) n 1 Z 0 L ( 1 = 2) m ( x 2 ; q ) L ( 1 = 2) n ( x 2 ; q ) dx E q ( x 2 ) = ( 1) m + n ( q ; q ) m ( q ; q ) n 1 Z 0 L ( 1 = 2) m ( y ; q ) L ( 1 = 2) n ( y ; q ) y 1 = 2 dy E q ( y ) = ( q ; q ) 2 n d n ( 1 = 2) Æ mn = q n ( q 1 = 2 ; q 1 = 2 ) 2 n ( q 1 = 2 ; q ) 1 = 2 Æ mn ; (3.2)
Boletn de la So iedad Mexiana de Matematias 8 (2002), 221{232 5 where we have used at the last step the equality ( q 1 = 2 ; q ) n ( q ; q ) n = ( q 1 = 2 ; q 1 = 2 ) 2 n and the standard relation (for a = q 1 = 2 and = 1 = 2) ( a ; q ) 1 ( aq ; q ) 1 = ( a ; q ) ; (3.3) whih is valid for an arbitrary omplex . In the same way we nd 1 Z 1 H 2 m +1 ( x ; q ) H 2 n +1 ( x ; q ) dx E q ( x 2 ) = 2( 1) m + n ( q ; q ) m ( q ; q ) n 1 Z 0 L (1 = 2) m ( x 2 ; q ) L (1 = 2) n ( x 2 ; q ) x 2 dx E q ( x 2 ) = 2( 1) m + n ( q ; q ) m ( q ; q ) n 1 Z 0 L (1 = 2) m ( y ; q ) L (1 = 2) n ( y ; q ) y 1 = 2 dy E q ( y ) = ( q ; q ) 2 n d n (1 = 2) Æ mn = q ( n +1 = 2) ( q 1 = 2 ; q 1 = 2 ) 2 n +1 ( q 1 = 2 ; q ) 1 = 2 Æ mn ; (3.4) where we have used the relation (1 q 1 = 2 )( q 3 = 2 ; q ) n = (1 q n +1 = 2 )( q 1 = 2 ; q ) n ; (3.5) whih is a straightforward onsequene of the general denition of the q -shifted fatorial ( a ; q ) n . Putting (3.2) and (3.4) together results in the orthogonality relation (3.1). The p ositivity of Jakson's q -exp onential funtion E q ( x 2 ) for x 2 R and q 2 (0 ; 1) is obvious from its denition (2.5): for it is represented as an innite sum of p ositive terms (or an innite pro dut of p ositive fators). This ompletes the pro of. To onlude this setion, we note the obvious fat that in the limit as q ! 1 the orthogonality relation (3.1) redues to the well-known orthogonality prop erty 1 Z 1 H m ( x ) H n ( x ) e x 2 dx = p 2 n n ! Æ mn (3.6) of the Hermite p olynomials H n ( x ) with resp et to the normal distribution e x 2 . This follows immediately from the limit relations (2.9) and (2.10), up on using the fat that lim q ! 1 E q ((1 q ) z ) = e z : (3.7) 4 Reurrene relation The easiest way to identify the sequene (2.8) ( that is, to nd its appropriate nihe in the Askey-sheme of q -p olynomial families [4 ℄ ), is to derive a three-term reurrene relation for it. Sine an arbitrary p olynomial p n ( x ) satises a reurrene relation of the form (see [21 , p. 19℄) ( a n x + b n ) p n ( x ) = p n +1 ( x ) + n p n 1 ( x ) ; n 0 ; (4.1) one an try to evaluate the right-hand side of (4.1) for p n ( x ) = H n ( x ; q ) and to nd then suh partiular o eÆient n that leads to the left-hand side of (4.1) with some a n and b n .
6 R. Alvarez-No darse et. al. On a q -extension of the Hermite polynomials H n ( x ) Before starting this derivation we note that in what follows it proves onvenient to use the following form L ( ) n ( x ; q ) = ( q +1 ; q ) n ( q ; q ) n n X k =0 q k ( k + ) ( q +1 ; q ) k n k q ( x ) k (4.2) of the q -Laguerre p olynomials L ( ) n ( x ; q ), whih omes from the rst line in denition (2.3), up on using the relation ( q n ; q ) k ( q ; q ) k = ( 1) k q k ( k 1) = 2 nk n k q : (4.3) Let us rst onsider the ase when n in (4.1) is even. Then from (2.8) and (2.3) we nd that H 2 n +1 ( x ; q ) + 2 n ( q ) H 2 n 1 ( x ; q ) = ( 1) n ( q 3 = 2 ; q ) n x n X k =0 q k ( k +1 = 2) ( q 3 = 2 ; q ) k n k q ( x 2 ) k + 2 n ( q ) ( 1) n 1 ( q 3 = 2 ; q ) n 1 x n 1 X k =0 q k ( k +1 = 2) ( q 3 = 2 ; q ) k n 1 k q ( x 2 ) k : (4.4) The next step is to employ the relation (3.5) in order to rewrite the quotient ( q 3 = 2 ; q ) n = ( q 3 = 2 ; q ) k from the rst term in (4.4) as ( q 3 = 2 ; q ) n ( q 3 = 2 ; q ) k = 1 q n +1 = 2 1 q k +1 = 2 ( q 1 = 2 ; q ) n ( q 1 = 2 ; q ) k : (4.5) In the seond term in (4.4) one an use the evident relation ( q 3 = 2 ; q ) n 1 = ( q 1 = 2 ; q ) n = (1 q 1 = 2 ) and the same formula (3.5) for the fator ( q 3 = 2 ; q ) k . Also, we reall the prop erty of the q - binomial o eÆient n 1 k q = 1 q n k 1 q n n k q : (4.6) Putting this all together, we obtain H 2 n +1 ( x ; q ) + 2 n ( q ) H 2 n 1 ( x ; q ) = ( 1) n ( q 1 = 2 ; q ) n x n X k =0 q k ( k +1 = 2) ( q 1 = 2 ; q ) k n k q ( x 2 ) k 1 q k +1 = 2 1 q n +1 = 2 2 n ( q ) 1 q n k 1 q n : (4.7) The right-hand side of (4.7) should math with H 2 n ( x ; q ) = ( 1) n ( q 1 = 2 ; q ) n n X k =0 q k ( k 1 = 2) ( q 1 = 2 ; q ) k n k q ( x 2 ) k ; (4.8) multiplied by a 2 n ( q ) x + b 2 n ( q ). This means that the o eÆient 2 n ( q ) an b e found from the equation 1 q n +1 = 2 2 n ( q ) 1 q n k 1 q n = d n ( q ) q k (1 q k +1 = 2 ) ; (4.9) where d n ( q ) is some k -indep endent fator. It is not diÆult to verify that the only solution of the equation (4.9) is 2 n ( q ) = 1 q n and d n ( q ) = q n . Thus H 2 n +1 ( x ; q ) + (1 q n ) H 2 n 1 ( x ; q ) = q n x H 2 n ( x ; q ) : (4.10)
Boletn de la So iedad Mexiana de Matematias 8 (2002), 221{232 7 Similarly, in the ase of an o dd n from (4.8) we have H 2 n +2 ( x ; q ) + 2 n +1 ( q ) H 2 n ( x ; q ) = ( 1) n +1 ( q 1 = 2 ; q ) n +1 n +1 X k =0 q k ( k 1 = 2) ( q 1 = 2 ; q ) k n + 1 k q ( x 2 ) k + 2 n +1 ( q )( 1) n ( q 1 = 2 ; q ) n n X k =0 q k ( k 1 = 2) ( q 1 = 2 ; q ) k n k q ( x 2 ) k : (4.11) In this ase it is even easier to nd the o eÆient 2 n +1 ( q ). Indeed, one will obtain from (4.11) an expression [ a 2 n +1 ( q ) x + b 2 n +1 ( q )℄ H 2 n +1 ( x ; q ) only if the two onstant terms in (4.11) with k = 0 anel eah other. This means that the 2 n +1 ( q ) should satisfy the equation ( q 1 = 2 ; q ) n +1 ( q 1 = 2 ; q ) n 2 n +1 ( q ) ( q 1 = 2 ; q ) n [1 q n +1 = 2 2 n +1 ( q )℄ = 0 : (4.12) Consequently, 2 n +1 ( q ) = 1 q n +1 = 2 and, therefore, by (4.11) we obtain H 2 n +2 ( x ; q ) + (1 q n +1 = 2 ) H 2 n ( x ; q ) = ( 1) n +1 ( q 1 = 2 ; q ) n +1 n +1 X k =0 q k ( k 1 = 2) ( q 1 = 2 ; q ) k ( n + 1 k q n k q ) ( x 2 ) k = ( 1) n ( q 3 = 2 ; q ) n x 2 n X m =0 q ( m +1)( m +1 = 2) ( q 3 = 2 ; q ) m ( n + 1 m + 1 q n m + 1 q ) ( x 2 ) m = ( 1) n q n +1 = 2 ( q 3 = 2 ; q ) n x 2 n X m =0 q m ( m +1 = 2 ( q 3 = 2 ; q ) m n m q ( x 2 ) m = q n +1 = 2 x H 2 n +1 ( x ; q ) ; (4.13) up on using the readily veried relation n + 1 m + 1 q n m + 1 q = q n m n m q (4.14) for the q -binomial o eÆient n k q . From (4.10) and (4.13) it thus follows that the q - p olynomials H n ( x ; q ) satisfy the three-term reurrene relation of the form H n +1 ( x ; q ) + (1 q n= 2 ) H n 1 ( x ; q ) = q n= 2 x H n ( x ; q ) : (4.15) An examination of the reurrene relation (4.15) reveals that the H n ( x ; q ) are related to the disrete q -Hermite p olynomials ~ h ( x ; q ) of typ e I I. Indeed, hange the base q ! q 2 in (4.15) and substitute into it H n ( x ; q 2 ) = q n ( n 1) = 2 ~ h n ( x ; q ) (4.16) to obtain the reurrene relation (1.8) for the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I. There are two immediate onsequenes of the formula (4.16). Firstly, ombining (4.16) with (2.8), one obtains the relationship ~ h 2 n ( x ; q ) = ( 1) n q n (2 n 1) ( q 2 ; q 2 ) n L ( 1 = 2) n ( x 2 ; q 2 ) ; ~ h 2 n +1 ( x ; q ) = ( 1) n q n (2 n +1) ( q 2 ; q 2 ) n x L (1 = 2) n ( x 2 ; q 2 ) ; (4.17)
8 R. Alvarez-No darse et. al. On a q -extension of the Hermite polynomials H n ( x ) b etween the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I and the q -Laguerre p olynomials (2.3) with = 1 = 2. Seondly, one an represent the q -p olynomials H n ( x ; q ), initially dened by (2.8), in the unied form ( f (1.7)) H n ( x ; q 2 ) = i n 2 0 q n ; i x | q ; q n ! = q n ( n 1) = 2 x n 2 1 q n ; q 1 n 0 q 2 ; q 2 =x 2 ! : (4.18) As a onsisteny hek, one may try to verify diretly that H n ( x ; q 2 ), dened by (4.18), do es really oinide with the expressions, given by (2.8) for even and o dd values of n , resp e- tively. This will lead to the two identities 2 1 q n ; y 0 q ; q n +1 = 2 ! = q n ( n 1) = 2 y n 2 1 q n ; q 1 = 2 n 0 q ; q =y ! ; 2 1 q n ; y 0 q ; q n +3 = 2 ! = q n ( n +1) = 2 y n 2 1 q n ; q ( n +1 = 2) 0 q ; q =y ! ; (4.19) b etween the terminating basi hyp ergeometri series 2 1 . The relations (4.19) are straightforward to verify by using the expansion ( x ; q ) k = k X j =0 q j ( j 1) = 2 k j q ( x ) j (4.20) of the q -shifted fatorial ( x ; q ) k in terms of p owers of the variable x . Yet another onsequene of the relation (4.16) an b e formulated as the following orollary of theorem (1) in setion 3. Corollary 2 The sequene of the disrete q -Hermite polynomials ~ h n ( x ; q ) of type II, dened by (1.7), satises the orthogonality relation 1 Z 1 ~ h m ( x ; q ) ~ h n ( x ; q ) dx E q 2 ( x 2 ) = q n 2 ( q ; q ) n ( q ; q 2 ) 1 = 2 Æ mn (4.21) on the whole real line R with respet to the ontinuous positive weight funtion w ( x ) = 1 =E q 2 ( x 2 ) . Pro of . Change the base q ! q 2 in (3.1) and then use the relation (4.16) to obtain (4.21). As we noted in the intro dution, the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I are known as p olynomials with the disrete orthogonality, supp orted on the p oints x = q k ; > 0 ; k 2 Z . The fat that they satisfy also the ontinuous orthogonality relation (4.21) do es not ontradit the general theory of orthogonal p olynomials [22 , 23℄. The p oint is that the Hamburger moment problem asso iated with f ~ h n ( x ; q ) g is indeterminate, and therefore they are orthogonal with resp et to an innite lass of weight funtions, b oth ontinuous and disrete ones [19 ℄, [24 ℄{[26 ℄. C.Berg studied in [27 ℄ some families of disrete solutions to indeterminate moment problems and showed how they an b e used to generate absolutely ontinuous solutions to the same moment problems. In partiular, C.Berg derived in [27℄ the ontinuous weight funtion w ( x ) = 1 =E q 2 ( x 2 ) for the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I and evaluated its moments. It should b e emphasized that in our approah the orthogonality relation (4.21) emerges as a simple onsequene of the ontinuous orthogonality relation (2.4) for the q -Laguerre p olynomials L ( 1 = 2) n ( x ; q ).
Boletn de la So iedad Mexiana de Matematias 8 (2002), 221{232 9 Observe that the ontinuous orthogonality relation (4.21) is useful for deriving some integrals, involving the q -exp onential funtion E q ( z ). We do not go into this question in depth but merely present one of them as an example. One of the generating funtions for the disrete q -Hermite p olynomials of typ e I I has the form (see [4 , p. 119℄) 1 X n =0 q n ( n 1) = 2 ( q ; q ) n ~ h n ( x ; q ) t n = E q ( xt ) E q 2 ( t 2 ) : (4.22) Multiply b oth sides of (4.21) by a onstant fator (i tq m= 2 ) m ( i q n= 2 ) n = ( q ; q ) m ( q ; q ) n and sum over indies m and n from zero to innity with the aid of the generating funtion (4.22) and the denition of another Jakson's q -exp onential funtion e q ( z ) := 1 ( z ; q ) 1 = 1 X n =0 z n ( q ; q ) n ; j z j < 1 : (4.23) This results in the following integral 1 Z 1 E q ( iq 1 = 2 xt ) E q ( iq 1 = 2 x ) E q 2 ( x 2 ) dx = ( q ; q 2 ) 1 = 2 e q ( t ) E q 2 ( q t 2 ) E q 2 ( q 2 ) ; j t j < 1 : (4.24) Sine lim q ! 1 e q ((1 q ) z ) = e z and the q -exp onential funtion E q ( z ) has the same limit (see (3.7)), the formula (4.24) is a q -extension of the well-known integral 1 Z 1 e 2 i ( x y ) t e t 2 dt = p e ( x y ) 2 ; whih reets the imp ortant prop erty of the normal distribution e x 2 of b eing its own Fourier transform. 5 Conluding remarks We have disussed in the preeding setions how one an onstrut the partiular p olynomial sequene of q -extensions of the Hermite p olynomials, either in terms of the q -Laguerre p olynomials L ( ) n ( x ; q ), = 1 = 2, or in terms of the disrete q -Hermite p olynomials ~ h n ( x ; q ) of typ e I I. The sequene so dened satises the ontinuous orthogonality on R with resp et to the p ositive weight funtion w ( x ) = 1 =E q ( x 2 ). It was shown that this orthogonality relation leads to an interesting integral, involving Jakson's q -exp onential funtions. It seems that the same approah an b e implemented for deriving a q -extension of the generalized Hermite p olynomials H ( ) n ( x ) [28 ℄{[30 ℄ with the ontinuous orthogonality prop erty (the ase of disrete orthogonality requires a dierent tehnique, see [31 ℄). This study is under way. Aknowledgments We are grateful to C.Berg for disussions and for having drawn our attention to the refer- ene [27 ℄. The researh of RAN has b een partially supp orted by the Ministerio de Cienias y Tenologa of Spain under the grant BFM-2000-0206-C04-02 , the Junta de Andalua under grant FQM-262 and the Europ ean pro jet INTAS-2000-272. The partiipation of NMA in this work has b een supp orted in part by the UNAM{DGAPA pro jet IN112300.