THEORY OF MULTIVARIABLE BESSEL FUNCTIONS AND ELLIPTIC MODULAR FUNCTIONS

Transcription

1 LE MATEMATICHE Vol LIII (1998) Fasc II pp THEORY OF MULTIVARIABLE BESSEL FUNCTIONS AND ELLIPTIC MODULAR FUNCTIONS G DATTOLI - ATORRE - S LORENZUTTA The theory of multivariable Bessel functions is exploited to establish further lins with the elliptic functions The starting point of the present investigations is the Fourier expansion of the theta functions which is used to derive an analogous expansion for the Jacobi functions (sndncn) in terms of multivariable Bessel functions which play the role of Fourier coefcients An important by product of the analysis is an unexpected lin with the elliptic modular functions 1 Introduction The theory of generalized Bessel function (GBF) has been reviewed in Ref [6] The importance of these functions stems from their wide use in application [10] [7] and from their implications in different elds of pure and applied mathematics ranging from the theory of generalized Hermite polynomials [2] to the theory of elliptic functions [1] [5] As to this last point it has been shown that [5] exponents of the Jacobi functions exhibit expansions of the Jacobi-Anger type in which the ordinary BF is replaced by an innite variable GBF This paper is addressed to a further investigation on the lin exixting between multivariable GBF and elliptic functions In particular we will prove that the Jacobi functions can be written in terms of a trigonometric series Entrato in Redazione il 30 ottobre 1998

2 388 G DATTOLI - ATORRE - S LORENZUTTA whose expansion coefcients are innite variable GBF We will also prove that these functions provide a natural basis for the expansion of the elliptic modular functions Before entering into the specic details of the problem we will review the main points of the multivariable GBF theory The few elements we will discuss in the following are both aimed at maing the paper self-contained and xing the notation we will exploit in the following A two variable one index GBF is specied by the following generating function (1) exp 1 x 1 t 1 + x 2 t t t 2 or equivalently by the following series (2) J n (x 1 x 2 ) + + J n 2 (x 1 )J (x 2 ) t n J n (x 1 x 2 ) 0 < t < The extension to more variables is accomplished rather straightforwardly and in fact (3) exp 3 J n (x 1 x 2 x 3 ) x s t s 1 2 t s + + t n J n (x 1 x 2 x 3 ) J n 3 (x 1 x 2 )J (x 3 ) In a similar way we can construct GBF with 4 5 m variables The recurrence relations of a m-variable GBF can be written as (4) J n {x s } m 1 1 J n s {x s } m 1 J n+s {x s } m 1 x s 2 m 2n J n {x s } m 1 sx s J n s {x s } m 1 + J n+s {x s } m 1 The extension to innite variables has been shown possible under the assumption that the series s x s be convergent ([9]) The lin of manyvariable GBFs with trigonometric series is almost natural since the following

3 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 389 generalized form of the Jacobi Anger expansion holds [9] (5) exp i x s sin(sθ) + e inθ J n {x s } 1 Modied forms of many variable GBF can be introduced as well using the following Jacobi-Anger expansion ( 1 ) (6) exp x s cos(sϕ) + e inϕ I n {x s } 1 which is valid under the same restriction on the {x s } as in the J -case Function I n {x s } 1 satises the recurrences (7) I n {x s } 1 1 I n s {x s } 1 x s 2 2nI n {x s } 1 + I n+s {x s } 1 sx s I n s {x s } 1 I n+s {x s } 1 A particular type of many-variable GBF which will be largely exploited in the following sections is dened below (8) exp m x 2s 1 t 2s 1 1 t 2s 1 + t n (0) J n {x 2s 1 } m 1 where the superscript (0) stands for odd In the case of m 2 the function (0) J n (x 1 x 3 ) is specied by the series (9a) (0) J n (x 1 x 3 ) + J n 3 (x 1 )J (x 3 ) and the extension to a larger number of variables is obvious ( 1 ) The I n {x s } m 1 functions are modied GBF of rst ind and play the same role as I n (x) in the one variable case

4 390 G DATTOLI - ATORRE - S LORENZUTTA Analogously one can dene the modied version (0) I n ({x 2s 1 } m ) of (0) J n ({x 2s 1 } m ) and then consider the relevant extensions to the innitevariable case denoted by (0) J n ({x 2s 1 } s 1 ) and (0) I n ({x 2s 1 } s 1 ) respectively for which the following Jacobi-Anger expansions hold true (9b) i e e x 2s 1 sin[(2s 1)ϕ] x 2s 1 cos[(2s 1)ϕ] + + e inϕ(0) J n ({x 2s 1 } s 1 ) e inϕ(0) I n ({x 2s 1 } s 1 ) under the assumption that the series (2s 1) x 2s 1 be convergent An interesting result to be immediately quoted is the possibility of exploiting functions of the above type to establish non-linear Jacobi-Anger expansions ie expansions relevant to the generating functions (10) F(x θ; n) e ix(sin θ)n G(x θ; n) e x(cos θ)n In the specic cases of n 3 we obtain (11) e x(cos θ)3 e x 4 [3 cos θ+cos(3θ)] e ix(sin θ)3 e i x 4 [3 sin θ sin(3θ)] + + e inθ(0) I n 3 4 x x 4 e inθ(0) J n 3 4 x x 4 After these few remars we are able to introduce the specic topic of the paper namely the lin between GBFs and elliptic functions of the Jacobi-type 2 Generalized Bessel functions and theta elliptic functions In a previous paper ([5]) it has been proved that the Jacobi functions sn u dn u and cn u are lined to the innite variable GBF by Jacobi-Anger lie

5 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 391 expansion In other words using the Fourier expansions of the elliptic functions for real z it can be shown that (12) e ixsn u e xcn u + + e xdn u e π x 2K e inz(0) J n yq m q 2m 1 e inz(0) I n + yq m 1/2 1 + q 2m 1 q m e 2inz I n y 1 + q 2m where (13) z πu 2K q exp π K y 2π K K x and K and i K are the quarter periods of the elliptic functions specied by the integrals (14) K K π/2 0 π/2 0 dϕ 1 2 sin 2 ϕ m 1 m 1 m 1 dϕ 1 2 sin 2 ϕ Expansion analogous to (12) can be derived for the theta-functions too It is to be noted that Eqs (12) can be viewed as trigonometric series associated to the exponents of the Jacobi functions In this paper we will derive trigonometric series of cn u dn u etc whose coefcients are provided by innite variable GBFs The starting point of our analysis is the following representation of thetafunctions in terms of innite products namely θ 1 (z) 2q 1/4 sin zg(q) π (1 2q 2n cos(2z) + q 4n ) θ 2 (z) 2q 1/4 cos zg(q) π (1 + 2q2n cos(2z) + q 4n ) (15) θ 3 (z) G(q) π (1 + 2q2n 1 cos(2z) + q 4n 2 ) θ 4 (z) G(q) π (1 2q 2n 1 cos(2z) + q 4n 2 ) G(q) π 1 q 2n

6 392 G DATTOLI - ATORRE - S LORENZUTTA Taing the logarithm of the rst of Eqs (15) we obtain θ 1 (z) (16) ln 2q 1/4 G(q) sin z ln 1 2q 2n cos(2z) + q 4n The rhs of the above equation can be cast in a more convenient form by means of the expansion [11] 1/2 (17) ln 1 2x cos y + x 2 which holds for m1 x m m cos(my) (18a) x < exp( Imy) or supposing x < 1 (18b) Re ln 1 < Imy < Re ln 1 x x Accordingly the n th term of the summation in Eq (16) writes (19) ln 1 2q 2n cos(2z) + q 4n 2 m1 q 2nm m cos(2mz) whose validity is limited to the region (20) πimτ < Imz < πimτ (q e iπτ ) Inserting therefore Eq (19) into Eq (16) we end up with θ 1 (z) (21) ln 2q 1/4 G(q) sin z 2 which according to Eq (20) holds for q 2n n(1 q 2n ) cos(2nz) (22) πimτ < Imz < πimτ

7 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 393 Using the same reasoning we are able to write θ4 (z) q n (23) ln 2 G(q) n(1 q 2n ) cos(2nz) whose validity is limited to the region (24) π 2 Imτ < Imz < π 2 mτ Finally exploiting the identities (25) we nd (26) θ 2 (z) ln 2q 1/4 G(q) cos z ln θ 2 (z) θ 1 (z + π/2) θ 3 (z) θ 4 (z + π/2) 2 θ3 (z) 2 G(q) ( 1) n q 2n cos(2nz) n 1 q2n ( 1) n n q n cos(2nz) 1 q2n According to the previous results for z real we can express the theta functions (ϑ i (z) ϑ i (z)τ) in terms of innite variable GBF as reported below (27) θ 1 (z) 2q 1/4 G(q) sin z θ 2 (z) 2q 1/4 G(q) cos z θ 3 (z) G(q) θ 4 (z) G(q) e 2inz q 2m I n 2 m(1 q 2m ) m 1 2( 1) e 2inz m+1 q 2m I n m(1 q 2m ) m 1 e 2inz I n 2( q)m m(1 q 2m ) m 1 e 2inz I n 2q m m(1 q 2m ) m 1 The above formulae provide the direct lin between GBF and theta elliptic functions In the next section we will show how similar expression can be obtained for sn u cn u dn u Jacobi functions

8 394 G DATTOLI - ATORRE - S LORENZUTTA 3 Generalized Bessel function and Jacobi elliptic functions The principal Jacobi function sn cn and dn are quotient of θ theta functions indeed [11] (28) sn u 1 θ 1 (z) θ 4 (z) cn u θ 2 (z) θ 4 (z) dn u θ 3(z) θ 4 (z) where z πu 2K Exploiting the previous relations for the logarithm of the theta-functions we get from Eqs (28) (29) 2q 1/4 ln(sn u) ln sin z + 2 ln(cn u) ln 2q 1/4 cos z + 2 ln(dn u) 1 2 ln( ) q n n 1 + q cos(2nz) n 1 q n n 1 + ( q) cos(2nz) n q 2n 1 cos(4n 2)z 1 q 4n 2 (2n 1) which for real z yield (30) sn u 2q1/4 + cn u 2q 1/4 dn u + 2 q m sin[(2n + 1)z]I n m 1 + q m 1 m + 2 q m cos[(2n + 1)z]I n m 1 + ( q) m 1 m e 2inz (0) I n 4 2m 1 q 2m 1 1 q 4m 2 m 1 Analogous expression relevant to the product of elliptic function can be found eeping eg the derivatives of both sides of Eqs (30) with respect to u

9 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 395 thus getting (31) cn u dn u πq1/4 K sn u dn u πq1/4 K sn u cn u iπ K (2n + 1) cos[(2n + 1)z] 2 q m I n m 1 + q m 1 m (2n + 1) sin[(2n + 1)z] 2 q m I n m 1 + ( q) m 1 m + ne 2inz(0) 4 q 2m 1 I n 2m 1 1 q m 1 4m 2 We can now specialize the above relations for particular values of u to get further interesting identities By setting u 0 in Eq (30) we nd (32a) 1 1/2q 1/ q m I n m 1 + ( q) m 1 m (0) I n 4 2m 1 q 2m 1 1 q 4m 2 m 1 In a similar way u K yields (32b) 1 2 1/2 q 1/ ( 1) n q m I n m 1 + q m 1 m ( 1) n (0) 4 q 2m 1 I n 2m 1 1 q m 1 4m 2 These last relations provide the basis for the lin between multivariable GBF and elliptic modular functions

10 396 G DATTOLI - ATORRE - S LORENZUTTA 4 Concluding remars In the previous sections we have analyzed the lin existing between GBF and Jacobi elliptic functions Further interesting relations will be discussed in these concluding remars We believe that an important by product of the already developed analysis is the possibility of expressing the so called elliptic modular functions (EMF) in series of GBF with innite variables The EMF are usually denoted by ([11] [8]) f (τ) θ 4 2 (0 τ) θ 4 3 (0 τ) 2 (33) g(τ) θ 4 4 (0 τ) θ 4 3 (0 τ) 2 h(τ) f (τ) g(τ) and satisfy the properties (34) f (τ + 2) f (τ) f (τ) + g(τ) 1 g(τ + 2) g(τ) f (τ + 1) h(τ) which follow from the fact that (35) q e iπτ e iπ(τ +2) e iπ(τ +1) The lin between EMF and GBF is readily obtained The rst of Eqs (32a) yields indeed (36) 1/2 4 h(τ) 2e i π 4 τ 2 e imπτ I n m 1 + ( 1) m e m 1 imπτ while Eqs (32b) provide the identities (37) 1/2 4 f (τ) 2e i π 4 τ 1/2 4 g(τ) 2 ( 1) n e imπτ I n m 1 + e m 1 imπτ ( 1) n (0) 4 e (2m 1)iπτ I n 2m 1 1 e m 1 2(2m 1)iπτ

11 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 397 By eeping the fourth power of both sides of Eqs (36) and (37) and by using the Jacobi-Anger expansion for the GBF we get (38) h(τ) 16e iπτ f (τ) 16e iπτ g(τ) 8 e imπτ I n m 1 + ( 1) m e m 1 imπτ 8 ( 1) n e imπτ I n m 1 + e m 1 imπτ ( 1) n(0) 16 e (2m 1)iπτ I n 2m 1 1 e m 1 2(2m 1)iπτ It is worth noting that the properties of the EMF can be directly inferred from the above series denition in fact (39a) iπ(τ +1) f (τ + 1) 16e 8 ( 1) n imπ(τ +1) e I n m 1 + e m 1 imπ(τ +1) 8 16e iπτ ( 1) n ( 1) m e imπτ I n m 1 + ( 1) m e m 1 imπτ and since (39b) I n ( 1) s x s ( 1) s 1 n I n {xs } s 1 we end up with (39c) f (τ + 1) 16e iπτ namely the last of Eqs (34) By noting that the last of eqs (32a) yields e imπτ 8 I n m 1 + ( 1) m e m 1 imπτ (40) 1 + (0) I n 4 2m 1 e (2m 1)iπτ 1 e (4m 2)iπτ m 1 it appears convenient to introduce the further EMF (41) l(τ) + (0) 16 e (2m 1)iπτ I n 2m 1 1 e m 1 (4m 2)iπτ

12 398 G DATTOLI - ATORRE - S LORENZUTTA and note that (42) l(τ + 1) g(τ) Furthermore from the Jacobi-Anger expansion of multivariable GBF it also follows that (43) h(τ) 16e iπτ 2 e imπτ exp 4 m 1 + ( 1) m e imπτ m1 and similarly for the other functions A more general treatment regarding EMF and GBF will be presented elsewhere Before closing this paper it is worth giving further identities which albeit an almost direct consequence of the considerations developed in the previous sections provide a deeper insight into the lin between GBF and elliptic functions By integrating both sides of Eqs (30) we obtain (44) (0) 4 q 2m 1 1 I n 2m 1 1 q m 1 4m 2 2K 2 q m I n m 1 + ( 1) m q m m 2 q 1/4 2K 4K 0 2K 0 1/2 πun i dn(u )e K du cn(u ) cos[(2n + 1) πu 2K ] du Analogous relations involving the theta functions can also be obtained thus getting eg (45) 1 I n {( 1)m q m } m 1 2π G(q) +π +π π θ 3 (z)e 2inz dz 1 I n { qm } m 1 θ 4 (z) cos(2nz) dz π G(q) 0 2q m q m m(1 q 2m ) This paper is a further contribution to the theory of GBF and to their lin with elliptic functions All the possible implications are far from being completely understood a forthcoming note will be devoted to a deeper insight

13 THEORY OF MULTIVARIABLE BESSEL FUNCTIONS 399 REFERENCES [1] P Appell Sur linversion approchée de certaines intégrals réelles et sur lextension de léquations du Kepler et des fonctions de Bessel CR Acad Sci 160 (1915) p 419 [2] P Appell - J Kampé de Fériet Fonctions Hypergéométriques et Hypersphériques Polynomes dhermite Gauthiers Villars Paris 1926 [3] LS Brown - TW Kibble Interactions of Intense Laser Beam with Electrons Phys Rev 133 (1964) p 705 [4] G Dattoli - C Chiccoli - S Lorenzutta - G Maino - A Torre Theory of Generalized Hermite Polynomials Computers Math Applic 28 (1994) p 71 and references therein [5] G Dattoli - C Chiccoli - S Lorenzutta - G Maino - M Richetta - A Torre Generating Functions of Multivariable Generalized Bessel Functions and Jacobi- Elliptic Functions J Math Phys 33 (1992) p 25 [6] G Dattoli - S Lorenzutta - G Maino - A Torre Generalized Forms of Bessel Functions and Hermite Polynomials Annals of Numerical Mathematics 2 (1995) p 211 [7] G Dattoli - A Renieri - A Torre Lectures on the Theory of Free Electron Laser and Related Topics World Scientic Singapore 1993 [8] ET Davis Introduction to the Theory of Non Linear Differential Equations Dover Pub New Yor 1962 [9] S Lorenzutta - G Maino - G Dattoli - A Torre - C Chiccoli On Innite-Variable Bessel Functions Rend Mat 15 (1995) p 405 [10] HR Reiss Absorption of Light by Light J Math Phys 3 (1962) p 59 [11] F Tricomi Funzioni Ellittiche Zanichelli Bologna 1951 G Dattoli - A Torre ENEA Dipartimento Innovazione Settore Fisica Applicata Centro Ricerche Frascati PO Box Frascati Roma (ITALY) S Lorenzutta ENEA Dipartimento Innovazione Settore Fisica Applicata Bologna (ITALY)