SYMMETRIC Q-BESSEL FUNCTIONS

Transcription

1 LE MATEMATICHE Vol. LI (1996) Fasc. I, pp SYMMETRIC Q-BESSEL FUNCTIONS GIUSEPPE DATTOLI - AMALIA TORRE analog of bessel functions, symmetric under the interchange of and 1 are introduced. The denition is based on the generating function realized as product of symmetric -eponential functions with appropriate arguments. Symmetric -Bessel function are shown to satisfy various identities as well as second-order -differential euations, which in the limit 1 reproduce those obeyed by the usual cylindrical Bessel functions. A brief discussion on the possible algebraic setting for symmetric -Bessel functions is also provided. 1. Introduction. Many special function of mathematical physics have been shown to admit generalizations to a base, which are usually reported as -special functions. Interest in such functions is motivated by the recent and increasing relevance of analysis, originally suggested almost a century ago [22], in eactly solvable models in statistical mechanics ([1],[23]). Like ordinary special functions, -analogs satisfy second order -differential euations and various identities or recurrence relations. Basic hypergeometric series are the prototype of -special functions, their properties and applications have been deeply investigated in ([9],[12]). Basic analogs of Bessel functions have been introduced by Jackson [13] and Swarthow [21] as -generalizations of the power series epansions, which Entrato in Redazione l11 settembre 1996.

2 154 GIUSEPPE DATTOLI - AMALIA TORRE denes the ordinary cylindrical Bessel functions. Three different types of such -etension can be recognized, each of them satisfy recurrence relations, second-order -differential euations and addition theorems, which reduce to those holding for the usual Bessel functions in the limit 1. In analogy with the usual special functions, -functions have recently been shown to admit an algebraic interpretation as matri elements of -eponentials of uantum-algebra generators on appropriate representation species. In [10], [11], for instance, a uantum-algebraic framework for -Bessel functions is provided as well as for the basic hypergeometric function. Finally, in [6] the generating function method is proposed as alternative unifying formalism, where the various -Bessel functions can be framed. The above uoted investigations use the method of the standard -analysis. The discovery of uantum groups and algebras ([14], [8], [24]), characterized by deformed communication relations, which generalize the canonical commutation relations, has led to interest in -analysis, which is symmetric under the interchange 1. Within that contet, a lot of attention is devoted to the so-called - oscillators ([2], [18]) and to their possible physical applications in such elds as atomic and nuclear physics [18], uantum optics [4] and superintegrable systems [5]. In this connection, it is natural to investigate the possibility of introducing -functions, which are symmetric under the interchange of and 1/. They are called symmetric to distinguish them from the standard ones. Symmetric -eponential and gamma functions have been etensively studied in [16], [17]. In this paper, we address the problem of dening symmetric -Bessel functions; in particular we follow the approach developed in [6] using indeed the symmetric -eponential function to realize the generating functions. Accordingly, in Section 2 we briey review the denition and the relevant properties of the symmetric -eponential function. In Section 3 symmetric - Bessel functions are dened and are shown to satisfy various identities, which, in the limit 1, reproduce the well-known recurrence relation obeyed by the usual cylindrical Bessel functions. In Section 4 we recognize the possibility of introducing shifting operators, which are then used to obtain the second-order -differential euations obeyed by the symmetric -Bessel functions. Finally, concluding comments on the possible algebraic setting for these functions as well as on the possible modied versions of them are given in Section 5.

3 SYMMETRIC Q-BESSEL FUNCTIONS Symmetric -eponential functions. Before entering the specic topic of the paper, let us briey review the properties of the -eponential functions, which will be basic to the forthcoming discussion on symmetric -Bessel functions. Denitions of functions in -analysis are borrowed from ordinary analysis through an appropriate generalization or -deformation. Accordingly, the -eponential function is introduced as eigenfunction of the -differentiation. Hence, since standard and symmetric differentiations can be introduced in -analysis, standard and symmetric -eponential functions are dened. For completenesssake, we report in the following the denition of standard - derivative, although for a detailed account of the relevant properties the reader is addressed to [16], [17]. Standard -derivatives is indeed dened as (1) d f (z) d z = f (z) f (z) ( 1)z which suggests to introducing the differentiation operator (2) ˆD = [( 1)z] 1 { zd/dz 1}. On the other hand, as noticed, in uantum groups signicant role is played by operators or functions, which are symmetric under the interchange of with 1/. Symmetric -derivative of an entire analytical function f (z), z C, is indeed introduced through the denition ([16], [17], [20]) (3) d d(z : ) f (z) = f (z) f ( 1 z) ( 1 )z which in analogy with the standard -differentiation suggests to introducing the operator (4) ˆD (z:) ( 2 1)z [ zd/dz zd/dz ]. The following link with the standard -differentiation operator ˆD is immediately recognized: (5) ˆD (z:) = + 1 [ ˆD + 1 ˆD 1/ ]

4 156 GIUSEPPE DATTOLI - AMALIA TORRE and the more direct relation with ˆD 2 can be stated (6) ˆD (z:) f (z) = ˆD 2 zd/dz f (z) = ˆD 2 f ( 1 z). It is also evident that the operator (5) is invariant with respect to the interchange 1/; eplicitly (7) ˆD (z:) = ˆD (z:1/). Let us briey review the main properties of the operator (5). Acting on powers of z, the operator ˆD (z:) gives (8) ˆD (z:) z n = [n] z n 1 where the symbol [n] denotes the number (9) [n] = n n 1 freuently occurring in the study of -deformed uantum mechanical simple harmonic oscillator where 0 < < 1 ([2], [18]). We list some relations satises by [n], since they will be used in the following. It is easy to prove that (10) [n] = [n] 1/ = [ n] [m + n] = n [m] + m [n] = m [n] + n [m] [0] = 0 [1] = 1. It is also useful to report the properties of the symmetric -differentiation, which satises a sum rule, a product rule and, in special cases, a chain rule as follows ˆD (z:) [ f (z) + g(z)] = D (z:) f (z) + ˆD (z:) g(z) ˆD (z:) [ f (z)g(z)] = g( 1 z) ˆD (z:) f (z) + f (z) ˆD (z:) g(z) = (11) = g(z) ˆD (z:) f (z) + f ( 1 z) ˆD (z:) g(z) ˆD (z:) [ f (αz)] = α ˆD (αz:) f (αz) ˆD (z:) f (z n ) = [n] z n 1 ˆD (zn : n ) f (z n ).

5 SYMMETRIC Q-BESSEL FUNCTIONS 157 As already noticed, -analogs of eponential functions are dened as eigenfunctions of differentiation operators. Conseuently, standard and symmetric -eponential functions can be introduced according to whether the operators ˆD or ˆD (z:) are considered. Limiting ourselves to the operator ˆD (z:), symmetric -eponential function E (z) can be introduced according to (12) ˆD (z:) E (z) = E (z) with furthermore E (z) being reuested to be regular at z = 0 and E(0) = 1. If C, = 0 and = 1, there is a uniue function satisfying the reuired conditions. Hence, since E 1/ (z) satises (12) as well, we have that E (z) = E 1/ (z), thus conrming that E is symmetric in the parameter with respect to the interchange of and 1. The power series epansion of E is given as (13) E (z) = n=0 z n [n]! with the -factorial [n]! being dened as 1 n = 0 (14) [n]! = n [r] n 1. r=1 The condition (12) is easily veried to hold according to the rule (8). The power series (13) has innite radius of convergence, and hence E (z) is an entire function for all = 0, = 1. It is worth stressing that the eponential function E (z) does not satisfy the semigroup property, i.e. E ()E (y) = E ( + y). Using indeed the power series epansion (13), we can write (15) E ()E (y) = r=0 r [r]! k=0 y k [k]!. An appropriate rehandling of the sums entering the above epression allows to recast the product E ()E (y) in the form (16) E ()E (y) = n=0 Z n (, y) [n]!

6 158 GIUSEPPE DATTOLI - AMALIA TORRE where Z n (, y) denotes the function (17) Z n (, y) = n h=0 n n h y h h and n can be understood as the -analog of the binomial coefcient: h (18) n h [n]! [h]![n h]!. However, the function Z n (, y) is not the -analog of binomial formula, giving the n-th power of the sum + y, which is indeed understood in the form (19) [ + y] n n h=0 n n h y h h(n+1). h Accordingly, the product (15) does not turn into -eponential of + y : E (z)e (y) = E ( + y). It might be interesting to derive the link between Z n and [ + y] n, which indeed reads (20) [ + y] n Z n(, y n+1 ). In addition, let us notice that the -binomial formula (19) satises the product rule: (21) [ + y] n [ + 2n y] m = [ + y]n+m and the following formulae for the -derivatives with respect to and y can be derived (22) ˆD (:) [ + y] n = [n] [ + y/] n 1 ˆD (y:) [ + y] n = [n] 2n [ + y] n 1 which give the usual formulae in the limit 1. Finally, it is worth stressing that the -analogue of the binomial formula is not symmetric in the parameter under the interchange 1. In the net section we use the symmetric -eponential function E (z) to generate symmetric -Bessel functions.

7 3. Symmetric -Bessel functions. SYMMETRIC Q-BESSEL FUNCTIONS 159 Let us consider the product of symmetric -eponential functions as (23) G (; t : ) = E t 2 E t 2 whose epression as series of t-powers can be easily obtained using the power series epansion of E given in (13). Eplicitly, we have (24) G (; t : ) = + t n n= k=0 ( 1) s (/2) n+2k [n + k]![k]!. Introducing symmetric -Bessel functions J n ( : ) in full analogy with the standard case [10], that is as the coefcients of the epansion (25) G (; t : ) + n= t n J n ( : ) it is immediate to get the eplicit epression of J n ( : ) as -power series (26) J n ( : ) = + k=0 ( 1) k (/2) n+2k [n + k]![k]!. It is evident that J n ( : ) is symmetric under the interchange 1 (27) J n ( : ) = J n ( : 1/). Using the relation (28) [n]! = (n) 2! 2/n(n 1) we can write (26) as (29) J n ( : ) = n2 /2 n2 /2 ( 1) k k=0 k=0 2 n+2k k(n+k) (k) 2!(n + k) 2! ( 1) k n+2k 2 k(n+k) (k) 1/ 2!(n + k) 1/ 2! =

8 160 GIUSEPPE DATTOLI - AMALIA TORRE where the symbol (n) means (30a) (n) = 1 n 1 and correspondingly (30b) (n)! = n (k). The functions (26) are characterized by recurrence relations, which can be easily obtained by taking the derivative of G with respect to. In fact, an account of the two possible epression for the symmetric -derivative of a product as reported in (11), we obtain the two relations involving the -derivative J n ( : ) ˆD (:) J n ( : ), and the contiguous functions J n 1 ( : ), J n+1 ( : ), namely k=1 (31) 2Jn n 1 ( : ) = 2 J n 1 : n+1 2 J n+1 ( : ) 2Jn n 1 ( : ) = 2 J n 1 ( : ) n+1 2 J n+1 : which can be obtained from each other by changing 1/. On the other hand, by multiplying the epression (26) by [n] and eploiting the property of [n], according to which (see E. (10)) the further relation can be inferred [n] = s [n + s] n+s [s] (32a) 2[n] J n ( : ) = (n 1) 2 J n 1 ( : ) + n+1 2 J n+1 ( : ) which by changing 1/ turns into (32b) 2 [n] J n ( : ) = n 1 2 J n 1 : + (n+1) 2 J n+1 :. Useful relations can be obtained by combining the above euations or directly eploiting the series epansion (26).

9 (33) SYMMETRIC Q-BESSEL FUNCTIONS 161 For instance, from (31) and (32) one gets n 1 2 J n : (n 1) 2 J n 1 ( : ) = = n+1 2 J n+1 ( : ) (n+1) 2 J n+1 : by putting / and in the above euation, the further relations follows 2 1 (34) [n] J n+1 ( : ) = 1 J n+1 : + n J n 1 : J n+1 ( : ) + n J n 1 ( : ) and (35) 2 1 [n] J n 1 ( : ) = J n 1 ( : ) + n J n+1 ( : ) 1 J n 1 : + n J n+1 :. Correspondingly, by using the epression (26) the following relations can be proved (n 1) 2 J n 1 : = 2 (1 2 )J n ( : ) + n 1 2 J n 1 : (36) (n+1) 2 J n+1 : = 2 (1 2 )J n+2 ( : ) + n+1 2 J n+1 : which state a further link between the contiguous functions J n ( : ), J n 1 ( : ) and J n+1 ( : ). It is worth stressing that there does not eist the analog for the ordinary functions. Let us not that in the relations (31) the functions J n 1 and J n+1 appear with different arguments / and respectively or viceversa. Conversely, in the relations (32) the functions J n 1 and J n+1 have the same argument, namely and /. Eploiting the relations (36), one can rewrite the basic recurrences (31) in a form symmetric with respect to the arguments of J n 1 and J n+1 ; and correspondingly one can turn (32) in a fashion where J n 1 and J n+1 have different arguments. Eplicitly, we have (37a) J n ( : ) = 2 ˆD (:) (1 2 ) 2 = n 1 2 J n 1 : (n+1) 2 J n+1 :

10 162 GIUSEPPE DATTOLI - AMALIA TORRE which changing into 1/ gives (37b) 2 ˆD (:) + (1 2 ) J n ( : ) = 2 = (n 1) 2 J n 1 : Similarly, combining (32) and (36) one gets (38a) 2[n] 2 (1 2 ) J n ( : ) = = (n 1) 2 J n 1 : n+1 2 J n+1 :. + n+1 2 J n+1 : (38b) 2[n] + 2 (1 2 ) J n ( : ) = = (n 1) 2 J n 1 : + n+1 2 J n+1 :. The brief discussion clearly display the wealth of possible relations, which can be drawn, involving J n ( : ), the contiguous functions J n 1 ( : ), J n+1 ( : ) and the derivative Jn ( : ). This is a direct conseuence of that the base differs from unity: = 1; in fact, as noticed in connection with (36), some of these relations do not have the analog in the limit 1, i.e. for the ordinary Bessel functions. 4. Shifting operators and differential euations. In the theory of ordinary Bessel functions the relevant recurrence relations allow to recognize shifting operators Ê and Ê +, which turn the functions of order n into the functions of the order n 1 and n + 1, respectively. They play a signicant role within the framework of group-theoretical interpretation of ordinary Bessel functions. In a purely mathematical contet they allow to derive the differential euation obeyed by the Bessel functions in a straightforward way. Similarly, shifting operators can be introduced for -Bessel functions as well, although in this case on turning the function of order n into that of order n 1 or n + 1 they also rescale the argument by the factor or 1/. Accordingly, four types of shifting operators can be dened. Indeed, combining

11 SYMMETRIC Q-BESSEL FUNCTIONS 163 appropriately the recurrence relations (32) and (37), it is easy to obtain (39) Ê (,n) () = n 1 [n] 2 Ê (,n) + () = (n+1) [n] 2 which act on J n ( : ) according to + ˆD (:) + (1 2 ) 4 ˆD (:) (1 2 ) 4 (40) Ê (,n) ()J n ( : ) = J n 1 ( : ) Ê (,n) + ()J n ( : ) = J n+1 ( : ) the argument of the functions J n 1 and J n+1 being rescaled by the factor. Correspondingly, the further operators Ê (1/,n) (), Ê (1/,n) + () eplicitly given by Ê (1/,n) () = (n 1) [n] 2 + ˆD (:) (1 2 ) 4 (41) Ê (1/,n) + () = n+1 [n] 2 ˆD (:) + (1 2 ) 4 can be recognized, which differ from the above (39) since the argument of the functions J n 1 and J n+1 is rescaled by 1/ ; namely (42) Ê (1/,n) ()J n ( : ) = J n 1 ( : ) Ê (1/,n) + ()J n ( : ) = J n+1 ( : ). It is needless to say that the operators (41) can be obtained from (39) by simply changing into 1/. However they are crucial in deriving the differential euation obeyed by J n ( : ). Indeed, it is evident the following identity (43) Ê (1/,n+1) ( )Ê (,n) + ()J n ( : ) = J n ( : ) where it has been eplicitly indicated that the operator Ê (,n) + acts on a function of order n and argument, whilst Ê (1/,n+1) acts on the function of order n + 1 and with rescaled argument.

12 164 GIUSEPPE DATTOLI - AMALIA TORRE Since (44) Ê (1,/,n+1) ( ) = n+1 2 [n + 1] + ˆD (:) 4 (1 2 ) after some algebra we end up with (45) Ẑ (n) ()J n ( : ) = 0 where the -Bessel operator Ẑ (n) has the rather complicated epression (46) Ẑ (n) () = [n]2 2 ( ) (n + n ) 1 2 [n] (1 + 2 )(1 2 ) 2 ( n + n ) 1 2 ˆD (:) ˆD 2 (:) n+1 which reduces to the ordinary Bessel operator in the limit 1. The above operator is not symmetric under the interchange of and 1, thus providing a further -differential euation obeyed by J n ( : ) (47) Ẑ (n) 1/ () = [n]2 2 2 (1 + 2 ) (1 2 )( n + n ) [n] (1 + 2 )(1 2 ) 2 ( n + n ) 1 2 ˆD (:) ˆD 2 (:) (n+1). Before closing the section, let us discuss the symmetric properties of J n ( : ) with respect to the inde n and the argument, multiplication and addition formulas being discussed in [23]. It is easy to prove that (48) J n ( : ) = ( 1) n J n ( : ) J n ( : ) = J n ( : ) strongly reminiscent of the corresponding relations obeyed by the usual Bessel functions.

13 5. Concluding remarks. SYMMETRIC Q-BESSEL FUNCTIONS 165 The analysis performed in the previous section conrms the possibility of introducing -analogs of Bessel function, which are symmetric in the parameter with respect to the interchange of and 1/. In full analogy with the ordinary case, recurrence relations, shifting operators and -differential euations can be obtained. The approach we follow is based on the generating function; which is dened as the product of symmetric -eponential functions with appropriate arguments. As already noticed, -Bessel functions linked to the standard -eponential function e (z) have been introduced ([13], [21], [6]); three different types of functions can be recognized in this case as a conseuence of the noninvariance of the eponential e with respect to the interchange 1. The properties of these functions have been deeply investigated, also within the contet of the theory of group representation. In [10],[11] it has been shown that -generalizations of Bessel functions appear in the realization of the twodimensional Euclidean uantum algebra E (2). Similarly, symmetric -Bessel functions can be understood as matri elements of the representation of E (2) on the Hilbert space of all the linear combinations of the functions z n, z C and n N. The group elements are realized as product of eponentials of the three generators, the symmetric -eponential function E being involved. In this connection, it is needless to say that the generating function method can be immediately recovered within the group representation framework, as shown in [7]. As a conclusion, let us stress that symmetric -analog of modied Bessel functions can be dened uite straightforwardly. In particular we note that they are specied by the generating function t (49) E E ( t ) = t n I n ( : ). The analytical continuation formula (50) i n I n ( : ) = J n (i : ) is a conseuence of the identity it (51) E E i 2 2t (it) = E E. 2 2(it) The -differential euation satised by I n ( : ) can be obtained from E. (47), by replacing with i.

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