EXPONENTIAL OF A SERIES

Transcription

1 JACKSON S -EXPONENTIAL AS THE EXPONENTIAL OF A SERIES arxiv:math-ph/ v1 2 May 2003 C. QUESNE Physiue Nucléaire Théoriue et Physiue Mathématiue, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium cuesne@ulb.ac.be Abstract Jacson s -exponential is expressed as the exponential of a series whose coefficients are obtained in closed form. Such a relation is used to derive some properties of the -exponential. Running head: Jacson s -Exponential Keywords: Basic special functions; -exponential PACS Nos.: Uw, Gp Since the advent of uantum groups, Jacson s -exponential 1 has found a lot of applications in physics. As an example, we may uote its use to describe some generalized coherent states in uantum optics 2, 3. Properties of the -exponential are often derived from its expression as an infinite product 4. For practical purposes, however, it may be useful to now the lin between the -exponential and an elementary function, such as the ordinary exponential. It is the purpose of the present letter to establish such a relation. Jacson s -exponential, defined by a E (z) 0 z!, (1) a Jacson has actually introduced two different inds of -exponentials, but the other one is related to the inverse of that considered here. 1

2 where and 1 if 0,! if 1,2,..., , (3) has a finite radius of convergence (1 ) 1 if 0 < < 1, but converges for all finite z if > 1 5. In the appropriate region of definition, let us try to express it as the exponential of some series, ( ) E (z) exp c ()z. (4) 1 This amounts to expanding in Taylor series the logarithm of the -exponential, (2) lne (z) c ()z. (5) 1 In Ref. 6, it has been shown that if the functions f(z) 0 a z and h(z) lnf(z) 1 c z, where a 0 1, are analytic in some neighbourhood of zero, then the Taylor coefficients of h(z) satisfy the recursion relation c a a c, 2,3,..., (6) with c 1 a 1. Applying this result to the logarithm of the -exponential leads to the relations 1 c ()! 1 1 1! c (), 2,3,..., (7) c 1 () 1. (8) It is straightforward to chec that the solution of E. (7), satisfying condition (8), is provided by c () (1 ) 1, 1,2,3,... (9) Inserting such an expression in E. (7) converts the latter into the relation 1 (1 ) 1 1!, 2,3,..., (10) 2

3 where!!! denotes a -binomial coefficient 5. Euation (10) can be easily proved by induction over by using the recursion relation We indeed obtain (1 ) 1 1! 1 1 (1 ) 1 1!+ (11), 1,2,..., 1. (12) 2 0 +(1 ) 1 1! (1 ) 1 1! (1 ) 1 (1 ) 1! 1 (1 ) 1 1!+1 1 (1 )! ( 1)+1, (13) where in the last step use has been made of the induction hypothesis. Euations (4) and (9) are the central result of this paper. To illustrate their usefulness, one can apply them to derive the following properties of the -exponential, already uoted in Ref. 4 (where we have corrected some misprints), E (z)e 1(z) 1, (14) ( ) 1 E (z)e ( z) E 2 1+ z2, (15) E (n z) n 1 m0 as well as a generalization of E. (15), ( n 1 (1 ) E (e 2πim/n n 1 z) E n n m0 E n( m z), n 2,3,..., (16) 3 z n ), n 2,3,... (17)

4 The proof of these relations is based upon the multiplicative property of the ordinary exponential, exp(x) exp(y) exp(x + y), and on some elementary properties of the coefficients c (), defined in (9), c ( 1 ) ( 1) 1 c (), (18) 2c 2 () ( ) 1 c ( 2 ), (19) 1+ n c ( n ) (n ) c (), (20) ( ) (1 ) n 1 nc n () c ( n ). n (21) We thin that the expression of Jacson s -exponential as the exponential of a series whose coefficients are nown in closed form may find some interesting physical applications. As a final point, it is worth noting that such a simple result does not seem to hold true for the -exponential defined in terms of symmetric -numbers. The author is indebted to K. A. Penson for informing her about Ref. 6 and for a valuable discussion. Thans are also due to R. Jagannathan and K. Srinivasa Rao for some interesting comments. The author is a Research Director of the National Fund for Scientific Research (FNRS), Belgium. 4

5 References 1 F. H. Jacson, Proc. Edin. Math. Soc. 22, 28 (1904); Quart. J. Pure Appl. Math. 41, 193 (1910). 2 M. Ari and D. D. Coon, J. Math. Phys. 17, 524 (1976); A. Jannussis, G. Brodimas, D. Sourlas and V. Zisis, Lett. Nuovo Cimento 30, 123 (1981); T. K. Kar and G. Ghosh, J. Phys. A29, 125 (1996). 3 C. Quesne, K. A. Penson and V. M. Tachu, Maths-type -deformed coherent states for > 1, uant-ph/ M. R. Ubriaco, Phys. Lett. A163, 1 (1992). 5 H. Exton, -Hypergeometric Functions and Applications (Ellis Horwood, Chichester, 1983). 6 M. Pourahmadi, Amer. Math. Monthly 91, 303 (1984). 5