A new class of sum rules for products of Bessel

A new class of sum rules for products of Bessel functions G. Bevilacqua,1, a) V. Biancalana,1 Y. Dancheva,1 L. Moi,1 and T. Mansour2, b) 1)

CNISM and Dipartimento di Fisica, Universit`a di Siena, Via Roma 56,

53100 Siena, Italy 2)

Department of Mathematics, University of Haifa, 31905 Haifa,

Israel (Dated: February 13, 2011)

We derive a new class of sum rules for products of the Bessel functions of first kind. Using standard algebraic manipulations we extend some of the well known properties of Jn . Some physical applications of the results are also discussed. A comparison with the Newberger[J. Math. Phys. 23 (1982) 1278] sum rules is performed on a typical example. PACS numbers: 02.30.Lt, 78.20.Bh

a)

Electronic mail: [email protected]

b)

Electronic mail: [email protected]

1

I.

INTRODUCTION Bessel functions of first kind (Jn ) are among the most ubiquitous special functions of

the mathematical physics. Their properties are described in several monographs1–4 , in the encyclopedic Abramowitz & Stegun5 as well as in every advanced mathematics textbook. The generating function identity 1 e (w− w ) ≡ z 2

∞ X

Jn (z) wn

(1)

n=−∞

is the key formula used in discussing the frequency modulation (FM) of the electric field of a laser. In fact a pure sinusoidal FM reads as

E(t) =


E0 i(ωL t+M sin(Ω t)) e + c.c. , 2

(2)

where ωL is the laser carrier frequency, Ω is the modulation frequency, M the modulation index and M Ω the modulation depth. Using (1) with w = ei Ω t one finds the well-known sidebands structure E(t) =

∞ E0 X Jn (M ) ei(ωL +nΩ)t + c.c. 2 n=−∞

(3)

In some applications, for instance studying the response of a two-level system to a FM field, one has to deal with expressions like ∞ X Jn (M )Jn−s (M ) , As = γ + i n Ω n=−∞

(4)

where γ represents a resonance linewidth. Even if the sum could be done exactly using the Newberger sum rule6 , the result is quite involved (the details are given in Section III A). Alternatively, in the physical interesting limit  = M Ω/γ < 1 (see below for more details), a geometric expansion gives " #  2 ∞ 1 X iΩ Ω n2 + O(3 ) As = Jn (M )Jn−s (M ) 1 − n − γ n=−∞ γ γ

(5)

reducing the problem to the evaluation of sums of the form Bk,s ≡

+∞ X

nk Jn (M )Jn−s (M )

n=−∞

2

(6)

for integer values of k ≥ 0 and s. To the authors’ best knowledge these sums are not addressed in the literature in the general case although special values are known, for instance B0,0 = 1 from 9.1.76 of [5]. Finding a closed expression for Bk,s is the main goal of the present work. The paper is organized as follows. In Section II the main results are derived and discussed, while in Section III some applications are pointed out. Finally Section IV contains the conclusions.

II.

RESULTS We start the derivation of the main result by rewriting (1) in a slight different form f (y, θ) ≡ e

i y sin θ

=

∞ X

Jn (y) ei n θ .

(7)

n=−∞

Deriving the left-hand side k times one gets f (k) (y, θ) ≡

∂ k f (y, θ) = pk (y, θ)f (y, θ) ∂θk

(8)

where pk (y, θ) is a polynomial in cos θ and sin θ and can be easily recast in the form pk (y, θ) =

∞ X

Ck,n (y) ei n θ ,

(9)

n=−∞

with Ck,n (y) polynomials in y. An explicit form for these coefficients is worked out in Appendix A. Putting all together one finds f

(k)

(y, θ) =

!

∞ X

∞ X

n=−∞

q=−∞

Ck,q (y) Jn−q (y)

ei n θ

(10)

On the other hand, deriving the right-hand side of (7) one obtains ∞ X

f (k) (y, θ) =

(i n)k Jn (y) ei n θ .

(11)

n=−∞

An explicit comparison of (11) with (10) is performed in Appendix B. Now consider the expression F (y1 , y2 , θ) ≡ f (k) (y1 , θ)f (y2 , θ). A first form can be obtained using (9) and (7) as follows F (y1 , y2 , θ) = pk (y1 , θ) ei (y1 +y2 ) sin θ ! ∞ X = Ck,n (y1 )ei n θ n=−∞

=

∞ X

! Jm (y1 + y2 )ei m θ

m=−∞

∞ X

∞ X

q=−∞

m=−∞

! Ck,q−m (y1 ) Jm (y1 + y2 ) ei q θ .

3

(12)

In the meantime an equivalent formula can be written using (11) ! ! ∞ ∞ X X F (y1 , y2 , θ) = (i n)k Jn (y1 ) ei n θ Jm (y2 )ei m θ n=−∞

=

m=−∞

!

∞ X

∞ X

q=−∞

n=−∞

(13)

(i n)k Jn (y1 ) Jq−n (y2 ) ei q θ ,

and, given the completeness of the set {ei q θ , q = 0, ±1, ±2, . . .}, the coefficients in both expansions must be equal ∞ X

Ck,q−m (y1 ) Jm (y1 + y2 ) =

m=−∞

∞ X

(i n)k Jn (y1 ) Jq−n (y2 ).

(14)

n=−∞

This is the central result of the paper and can be considered as a generalization of the Jn addition formula [5, formula 9.1.75], which, in fact, is obtained for k = 0 noticing that C0,n = δn,0 (see Appendix A). Besides, with a proper choice of y1 and y2 different sum rules can be obtained. In fact, substituting y1 = −y2 = y in (14) and remembering that Jn (0) = δn,0 one finds Ck,q (y) = = =

∞ X

(i n)k Jn (y) Jq−n (−y)

n=−∞ ∞ X

(i n)k Jn (y) (−1)q−n Jq−n (y)

(15)

n=−∞ ∞ X

(i n)k Jn (y) Jn−q (y).

n=−∞

This gives an answer for the initial problem, namely Bk,s =

1 Ck,s ik

(16)

and the Bk,s for the first k values can be read from Appendix A. The companion case y1 = y2 = y is worked out with similar algebraic elaborations leading to the formula ∞ X

(−1)n nk Jn (y) Jn−q (y) =

n=−∞

A.

∞ (−1)q X Ck,q−m (y) Jm (2y). ik m=−∞

(17)

Generalizations Many generalizations are possible once one dives in the world of generalized Bessel

functions4 . Let us discuss some cases. 4

Consider the function e

i (x cos θ+y sin θ)

∞ X

≡

J˜n (x, y) ei n θ ,

(18)

n=−∞

which arises in the context of general sinusoidal modulation. The J˜n (x, y) are a kind of “generalized” Bessel functions, which can be given in terms of usual Bessel functions by means of simple algebraic manipulations (see also equation (11.10) of Dattoli4 ) ∞ X

J˜n (x, y) ≡

iq Jq (x) Jn−q (y).

(19)

q=−∞

The same steps as before lead to the sum rule ∞ h i∗ X n J˜n (x, y) J˜n−q (x, y) ≡ (y + i x) δq,1 + (y − i x) δq,−1 . 2

(20)

n=−∞

Furthermore k-order derivatives lead to sum rules with nk which the interested reader can easily work out. The second generalization is achieved considering e

i (y1 sin θ+y2 sin 2θ)

∞ X

≡

Jbn (y1 , y2 ) ei n θ .

(21)

n=−∞

These Jbn (y1 , y2 ) “generalized” Bessel functions can be expressed as ∞ X

Jbn (y1 , y2 ) ≡

Jq (y2 ) Jn−2 q (y1 ),

(22)

q=−∞

and are real function for real values of yi (see chapter 2 of Dattoli4 for further details). By means of the same device one gets ∞ X

n Jbn (y1 , y2 )Jbn−s (y1 , y2 ) ≡

n=−∞

y1 (δs,1 + δs,−1 ) + y2 (δs,2 + δs,−2 ). 2

(23)

Also in this case higher derivatives lead to other sum rules. In a complete general setting, one could consider the problem of a FM signal with no special properties, i.e. e

i φ(t)

=

∞ X

G n ei n Ω t

(24)

n=−∞

where the only requirement is that φ(t) is a real and periodic well-behaved function (a more precise statement is given below) φ(t) =

∞ X n=−∞

5

φn ei n Ω t .

(25)

The Gn coefficients are related in a very complicated way to multiple sums of products of ordinary Bessel functions, but the explicit form is not important here. They satisfy the “conservation of energy” sum rule ∞ X

Gn G∗n−s = δs,0 .

(26)

n=−∞

Taking the time derivative of (24) and doing the same steps as above we get ∞ X

n Gn G∗n−s = i s φs .

(27)

n=−∞

˙ Equation (27) holds provided that φ(t) can be expanded as a meaningful Fourier series, i.e. ˙ = φ(t)

∞ X

i n Ω φn ei n Ω t

(28)

n=−∞

must be convergent. As before, higher order derivatives will give higher order sum rules.

III.

APPLICATIONS

An example is considered where formula (16) comes handy. Let us model an optical resonance as a damped harmonic oscillator of unit mass, forced by the electric field of the light beam z¨ + γ z˙ + ω02 z = f e+iωt ,

(29)

where, as usual, the physical oscillator displacement is x(t) =