theory of multivariable bessel functions and elliptic ... - Le Matematiche

LE MATEMATICHE Vol. LIII (1998) Fasc. II, pp. 387399

THEORY OF MULTIVARIABLE BESSEL FUNCTIONS AND ELLIPTIC MODULAR FUNCTIONS G. DATTOLI - A.TORRE - S. LORENZUTTA The theory of multivariable Bessel functions is exploited to establish further links 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 (sn,dn,cn...) in terms of multivariable Bessel functions, which play the role of Fourier coef�cients. An important by product of the analysis is an unexpected link 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 B.F. is replaced by an in�nite variable GBF. This paper is addressed to a further investigation on the link 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.

388

G. DATTOLI - A.TORRE - S. LORENZUTTA

whose expansion coef�cients are in�nite 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 speci�c 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 making the paper self-contained and �xing the notation we will exploit in the following. A two variable one index GBF is speci�ed by the following generating function � � +∞ � � 1 �� 1� 1� � 2 x1 t − + x2 t − 2 t n Jn (x 1 , x 2), 0 < |t| < ∞ = (1) exp 2 t t n=−∞ or equivalently by the following series Jn (x 1 , x 2 ) =

(2)

+∞ �

Jn−2� (x 1 )J�(x 2 ).

�=−∞

The extension to more variables is accomplished rather straightforwardly and in fact

(3)

� 3 +∞ � � 1� xs � s t − s t n Jn (x 1 , x 2, x 3 ), = exp 2 t n=−∞ s=1 �

Jn (x 1 , x 2, x 3 ) =

+∞ �

Jn−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)

� � 1� � � � �� ∂ m Jn−s {x s }m Jn {x s }m = , 1 1 − Jn+s {x s }1 ∂ xs 2 m � � � � � � � �� m m sx s Jn−s {x s }m . 2n Jn {x s }1 = 1 + Jn+s {x s }1 s=1

The extension to in�nite variables has been shown possible, under the ∞ � s|x s | be convergent ([9]). The link of manyassumption that the series s=1

variable GBFs with trigonometric series is almost natural, since the following

389

THEORY OF MULTIVARIABLE BESSEL FUNCTIONS. . .

generalized form of the Jacobi Anger expansion holds [9] �

exp i

(5)

∞ � s=1

�

x s sin(sθ ) =

+∞ �

n=−∞

� � einθ Jn {x s }∞ 1 .

Modi�ed forms of many variable GBF can be introduced as well using the following Jacobi-Anger expansion (1 ) exp

(6)

�

∞ � s=1

�

x s cos(sϕ) =

+∞ �

e

inϕ

n=−∞

�

In {x s }∞ 1

�

,

which under the same restriction on the {x s } as in the J -case. Function � is valid � ∞ In {x s }1 satis�es the recurrences

(7)

� 1� � � � �� ∂ � ∞ ∞ I In {x s }∞ } } = {x + I {x , n−s s n+s s 1 1 1 ∂ xs 2 ∞ � � � � � � � �� ∞ ∞ sx } } = I {x − I {x . 2n In {x s }∞ s n−s s n+s s 1 1 1 s=1

A particular type of many-variable GBF which will be largely exploited in the following sections is de�ned below (8)

exp

�

m �

�

x 2s−1 t

s=1

2s−1

−

1 �

t 2s−1

�

=

+∞ �

n=−∞

� � t n (0) Jn {x 2s−1 }m 1 ,

where the superscript (0) stands for odd. In the case of m = 2 the function (0) Jn (x 1 , x 3) is speci�ed by the series (0)

(9a)

Jn (x 1, x 3 ) =

+∞ �

Jn−3� (x 1 )J� (x 3)

�=−∞

and the extension to a larger number of variables is obvious. �

�

(1 ) The In {xs }m 1 functions are modi�ed GBF of �rst kind and play the same role as In (x) in the one variable case.

390


Analogously, one can de�ne the modi�ed version, (0) In ({x 2s−1}m s=1 ) of m (0) Jn ({x 2s−1 }s=1 ) and then consider the relevant extensions to the in�nitevariable case denoted by (0) Jn ({x 2s−1 }s≥1 ) and (0) In ({x 2s−1}s≥1 ), respectively, for which the following Jacobi-Anger expansions hold true i

e (9b)

∞ �

x2s−1 sin[(2s−1)ϕ]

=

s=1

∞

�

x2s−1 cos[(2s−1)ϕ]

e s=1

=

under the assumption that the series

+∞ �

einϕ(0) Jn ({x 2s−1 }s≥1 ),

+∞ �

einϕ(0) In ({x 2s−1 }s≥1 ),

n=−∞

n=−∞ ∞ �

s=1

(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, i.e. expansions relevant to the generating functions n

F(x , θ ; n) = eix(sin θ) ,

(10)

n

G(x , θ ; n) = e x(cos θ) . In the speci�c cases of n = 3 we obtain 3

x

e x(cos θ) = e 4 [3 cos θ+cos(3θ)] = (11) 3

x

eix(sin θ) = ei 4 [3 sin θ−sin(3θ)] =

+∞ �

n=−∞ +∞ �

n=−∞

einθ(0) In

�3

x,

x� , 4

�3

x,

x� , 4

einθ(0) Jn

4

4

After these few remarks we are able to introduce the speci�c topic of the paper namely the link 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 linked to the in�nite variable GBF by Jacobi-Anger like


391

expansion. In other words, using the Fourier expansions of the elliptic functions, for real z it can be shown that �� 1 +∞ � yq m− 2 ixsn u inz(0) = e Jn , e 1 − q 2m−1 n=−∞ m≥1 �� +∞ � yq m−1/2 einz(0) In , e xcn u = (12) 1 + q 2m−1 n=−∞ m≥1 �� +∞ � πx qm xdn u 2inz = e 2K e In ky , e 2m 1 + q n=−∞ m≥1

where

� πK�� 2π πu , q = exp − , y= x 2K K kK and K and i K � are the quarter periods of the elliptic functions, speci�ed by the integrals � π/2 dϕ � , K = 0 1 − k 2 sin2 ϕ (14) � π/2 dϕ � � , k 2 + k � 2 = 1. K = 0 1 − k �2 sin2 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 coef�cients are provided by in�nite variable GBFs. The starting point of our analysis is the following representation of thetafunctions in terms of in�nite products, namely z=

(13)

∞

θ1 (z) = 2q 1/4 sin zG(q) π (1 − 2q 2n cos(2z) + q 4n ), n=1

∞

θ2 (z) = 2q 1/4 cos zG(q) π (1 + 2q 2n cos(2z) + q 4n ), n=1

∞

θ3 (z) = G(q) π (1 + 2q 2n−1 cos(2z) + q 4n−2 ),

(15)

n=1 ∞

θ4 (z) = G(q) π (1 − 2q 2n−1 cos(2z) + q 4n−2 ), n=1

�

�� ∞ � G(q) = π 1 − q 2n . n=1

392


Taking the logarithm of the �rst of Eqs. (15) we obtain ln

(16)

�

∞

� � � � θ1 (z) = ln 1 − 2q 2n cos(2z) + q 4n . 1/4 2q G(q) sin z n=1

The r.h.s. of the above equation can be cast in a more convenient form by means of the expansion [11] ln

(17)

��

1 − 2x cos y + x 2

�−1/2 �

=

∞ � xm cos(my) m m=1

which holds for |x | < exp(−Imy)

(18a) or supposing |x | < 1

� 1� � 1� −Re ln < Imy < Re ln . x x

(18b)

Accordingly, the n th term of the summation in Eq. (16) writes (19)

∞ � � � q 2nm cos(2mz) ln 1 − 2q 2n cos(2z) + q 4n = −2 m m=1

whose validity is limited to the region (20)

−π Imτ < Imz < π Imτ (q = eiπτ ).

Inserting therefore Eq. (19) into Eq. (16) we end up with (21)

ln

�

∞ � � q 2n θ1 (z) = −2 cos(2nz) 2q 1/4 G(q) sin z n(1 − q 2n ) n=1

which, according to Eq. (20), holds for (22)

−π Imτ < Imz < π Imτ.

393


Using the same reasoning we are able to write ∞ � θ (z) � � qn 4 = −2 cos(2nz) (23) ln 2n ) G(q) n(1 − q n=1 whose validity is limited to the region π π (24) − Imτ < Imz < mτ. 2 2 Finally exploiting the identities θ2 (z) = θ1 (z + π/2),

(25)

θ3 (z) = θ4 (z + π/2),

we �nd ln (26)

�

∞

� � (−1)n q 2n θ2 (z) = −2 cos(2nz), 2q 1/4 G(q) cos z n 1 − q 2n n=1

∞ � θ (z) � � (−1)n q n 3 = −2 cos(2nz). ln G(q) n 1 − q 2n n=1

According to the previous results, for z real we can express the theta functions (ϑi (z) = ϑi (z)τ ) in terms of in�nite variable GBF as reported below θ1 (z) = 2q 1/4 G(q) sin z θ2 (z) = 2q

1/4

(27) θ3 (z) = G(q) θ4 (z) = G(q)

G(q) cos z

+∞ �

e2inz In

n=−∞ +∞ �

e2inz In

n=−∞ +∞ �

n=−∞

−2

� � q 2m , m(1 − q 2m ) m≥1

�� 2(−1)m+1 q 2m � m(1 − q 2m )

e2inz In

��

−

2(−q)m � � , m(1 − q 2m ) m≥1

e2inz In

��

−

� � 2q m . m(1 − q 2m ) m≥1

n=−∞ +∞ �

��

m≥1

�

,

The above formulae provide the direct link 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.

394


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)

1 θ1 (z) , sn u = √ k θ4 (z)

cn u =

�

k � θ2 (z) , k θ4 (z)

dn u =

√ θ3 (z) , k� θ4 (z)

πu where z = 2K . Exploiting the previous relations for the logarithm of the theta-functions, we get from Eqs. (28)

∞ � � 2q 1/4 � 1 qn √ sin z + 2 cos(2nz), n 1 + qn k n=1 � ∞ � � � � qn 1 1/4 k cos z + 2 cos(2nz), ln(cn u) = ln 2q k n 1 + (−q)n

ln(sn u) = ln

(29)

n=1

ln(dn u) =

∞ �

q 2n−1 cos(4n − 2)z 1 ln(k � ) + 4 2 1 − q 4n−2 (2n − 1) n=1

which for real z yield

(30)

+∞ �� 2 q m � � 2q 1/4 � sin[(2n + 1)z]In , sn u = √ m 1 + q m m≥1 k n=−∞ � +∞ �� 2 � � � � qm 1/4 k cos[(2n + 1)z]In , cn u = 2q k n=−∞ m 1 + (−q)m m≥1

dn u =

+∞ �� √ � k� e2inz (0) In n=−∞

q 2m−1 � � 4 . 2m − 1 1 − q 4m−2 m≥1

Analogous expression relevant to the product of elliptic function can be found keeping e.g. the derivatives of both sides of Eqs. (30) with respect to u,


395

thus getting

cn u dn u =

+∞ π q 1/4 � √ (2n + 1) cos[(2n + 1)z]· K k n=−∞ �� 2 q m � � , · In m 1 + q m m≥1

+∞ π q 1/4 � (2n + 1) sin[(2n + 1)z]· (31) sn u dn u = K n=−∞

· In

� � qm , m 1 + (−q)m m≥1

�� 2

√ +∞ �� 4 q 2m−1 � � iπ k � � ne2inz(0) In . − k sn u cn u = K n=−∞ 2m − 1 1 − q 4m−2 m≥1 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)

+∞ �� 2 � � � qm 1 � k �1/2 −1/4 q = I , n m m≥1 2 k� m 1 + (−q) n=−∞ +∞ �� 4 � q 2m−1 � � 1 (0) In . √ = 2m − 1 1 − q 4m−2 m≥1 k� n=−∞

In a similar way, u = K yields +∞ �� 2 q m � � � 1 1/2 −1/4 k q = (−1)n In , 2 m 1 + q m m≥1 n=−∞

(32b) √

k� =

+∞ �

n=−∞

(−1)n (0) In

��

q 2m−1 � � 4 . 2m − 1 1 − q 4m−2 m≥1

These last relations provide the basis for the link between multivariable GBF and elliptic modular functions.

396


4. Concluding remarks. In the previous sections we have analyzed the link existing between GBF and Jacobi elliptic functions. Further interesting relations will be discussed in these concluding remarks. 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 in�nite variables. The EMF are usually denoted by ([11], [8])

(33)

f (τ ) =

θ24 (0|τ ) = k2 , θ34 (0|τ )

g(τ ) =

θ44 (0|τ ) = k� 2, θ34 (0|τ ) k2 k2 f (τ ) = − �2 = − , g(τ ) k 1 − k2

h(τ ) = − and satisfy the properties

(34)

f (τ + 2) = f (τ ),

g(τ + 2) = g(τ ),

f (τ ) + g(τ ) = 1,

f (τ + 1) = h(τ )

which follow from the fact that k 2 + k �2 = 1,

(35)

q = eiπτ = eiπ(τ +2) = −eiπ(τ +1) .

The link between EMF and GBF is readily obtained. The �rst of Eqs. (32a) yields indeed (36)

� k �1/2 k�

=

∞ �� 2 � � � � eimπτ π 4 −h(τ ) = 2ei 4 τ In , m 1 + (−1)m eimπτ m≥1 n=−∞

while Eqs. (32b) provide the identities k 1/2 = (37) k �1/2 =

� 4

� 4

π

f (τ ) = 2ei 4 τ

g(τ ) =

∞ �

n=−∞

∞ �

(−1)n In

n=−∞

(−1)n (0) In

��

eimπτ � � , m 1 + eimπτ m≥1

�� 2

e(2m−1)iπτ � � 4 . 2m − 1 1 − e2(2m−1)iπτ m≥1


397

By keeping the fourth power of both sides of Eqs. (36) and (37) and by using the Jacobi-Anger expansion for the GBF we get h(τ ) = −16eiπτ f (τ ) = 16eiπτ

(38)

g(τ ) =

∞ �

∞ �

In

n=−∞

∞ �

� � eimπτ , m 1 + (−1)m eimπτ m≥1

�� 8

(−1)n In

n=−∞

��

(−1)n(0) In

n=−∞

eimπτ � � , m 1 + eimπτ m≥1

�� 8

e(2m−1)iπτ � � 16 . 2m − 1 1 − e2(2m−1)iπτ m≥1

It is worth noting that the properties of the EMF can be directly inferred from the above series de�nition, in fact (39a)

f (τ + 1) = 16e = −16eiπτ

∞ �

iπ(τ +1)

(−1)n In

n=−∞ ∞ �

(−1)n In

n=−∞

eimπ(τ +1) � � m 1 + eimπ(τ +1) m≥1

�� 8

(−1)m eimπτ � � m 1 + (−1)m eimπτ m≥1

�� 8

and since In

(39b) we end up with (39c)

��

(−1)s x s

�

s≥1

f (τ + 1) = −16eiπτ

�

� � = (−1)n In {x s }s≥1

∞ �

n=−∞

In

� � eimπτ m 1 + (−1)m eimπτ m≥1

�� 8

namely, the last of Eqs. (34). By noting that, the last of eqs. (32a) yields (40)

+∞ �� 4 � e(2m−1)iπτ � � 1 (0) √ = In 2m − 1 1 − e(4m−2)iπτ m≥1 k� n=−∞

it appears convenient to introduce the further EMF (41)

l(τ ) =

+∞ �

n=−∞

(0)

In

��

e(2m−1)iπτ � � 16 2m − 1 1 − e(4m−2)iπτ m≥1

398


and note that (42)

l(τ + 1) = g(τ ).

Furthermore, from the Jacobi-Anger expansion of multivariable GBF it also follows that ∞ � � � eimπτ 2 (43) h(τ ) = −16eiπτ exp 4 m 1 + (−1)m eimπτ m=1

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 link between GBF and elliptic functions. By integrating both sides of Eqs. (30) we obtain � 2K �� 4 q 2m−1 � � π un 1 (0) In dn(u, k)e−i K du, = √ � 2m − 1 1 − q 4m−2 m≥1 2K k 0 �� 2 � � m q 1 −1/4 1 � k �1/2 (44) In q · = m 1 + (−1)m q m m≥1 2 2K k � � 4K πu ] du. cn(u, k) cos[(2n + 1) · 2K 0 Analogous relations involving the theta functions can also be obtained, thus getting e.g. � +π � � 1 m θ3 (z)e−2inz dz, In {(−1) q˜m }m≥1 = 2π G(q) −π � +π � � 1 (45) θ4 (z) cos(2nz) dz, In {q˜m }m≥1 = 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 link with elliptic functions. All the possible implications are far from being completely understood, a forthcoming note will be devoted to a deeper insight.


399

REFERENCES [1]

[2] [3] [4]

[5]

[6]

[7] [8] [9] [10] [11]

P. Appell, Sur linversion approchée de certaines intégrals réelles et sur lextension de léquations du Kepler et des fonctions de Bessel, C.R. Acad. Sci., 160 (1915), p. 419. P. Appell - J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques, Polynomes dHermite, Gauthiers Villars, Paris, 1926. L.S. Brown - T.W. Kibble, Interactions of Intense Laser Beam with Electrons, Phys. Rev., 133 (1964), p. 705. 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. G. Dattoli - C. Chiccoli - S. Lorenzutta - G. Maino - M. Richetta - A. Torre, Generating Functions of Multivariable Generalized Bessel Functions and JacobiElliptic Functions, J. Math. Phys., 33 (1992), p. 25. G. Dattoli - S. Lorenzutta - G. Maino - A. Torre, Generalized Forms of Bessel Functions and Hermite Polynomials, Annals of Numerical Mathematics, 2 (1995), p. 211. G. Dattoli - A. Renieri - A. Torre, Lectures on the Theory of Free Electron Laser and Related Topics, World Scienti�c, Singapore, 1993. E.T. Davis, Introduction to the Theory of Non Linear Differential Equations, Dover Pub., New York, 1962. S. Lorenzutta - G. Maino - G. Dattoli - A. Torre - C. Chiccoli, On In�nite-Variable Bessel Functions, Rend. Mat., 15 (1995), p. 405. H.R. Reiss, Absorption of Light by Light, J. Math. Phys., 3 (1962), p. 59. F. Tricomi, Funzioni Ellittiche, Zanichelli, Bologna, 1951.

G. Dattoli - A. Torre, ENEA, Dipartimento Innovazione, Settore Fisica Applicata, Centro Ricerche Frascati, P.O. Box 65, 00044 Frascati, Roma (ITALY) S. Lorenzutta, ENEA, Dipartimento Innovazione, Settore Fisica Applicata, Bologna (ITALY)