On the common zeros of Bessel functions

Journal of Computational and Applied Mathematics 153 (2003) 387 – 393

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On the common zeros of Bessel functions Eugenia N. Petropoulou∗;1 , Panayiotis D. Siafarikas, Ioannis D. Stabolas1 Department of Mathematics, University of Patras, Patras, Greece Received 15 October 2001; received in revised form 13 March 2002

Abstract Using a functional analytic method we give some results concerning common zeros of the ordinary Bessel functions J (z) of 5rst kind, with respect to  and 5xed z. A lower bound for the common zero z of the Bessel functions J (z), J (z), where ,  are known, is also given. c 2002 Elsevier Science B.V. All rights reserved.
MSC: 33C10 Keywords: Common zeros; Bessel functions

1. Introduction Investigating the common zeros of Bessel functions of 5rst kind, J (z), is a quite old and important problem of Bessel functions for which various theoretical and numerical results exist. One such result is that the Bessel functions J (z) and J (z), where  rational and  −  is a positive integer, cannot have common zeros (Bourget’s hypothesis) [14, p. 484]. A simple alternative proof of a speci5c case of this result was given in [9, p. 464]. However, this result does not exclude the possibility that two functions J (z) and J (z), where ;  do not di?er by a positive integer, may have common zeros. The problem of locating the zeros can be “attacked” in two ways. One way is to consider a 5xed z0 and investigate the function J (z0 ) as a function of . The 5rst who did this was Dougall in 1900 [4] who proved that, if z0 is purely imaginary, then the real part of  cannot be nonnegative. Later, in 1936, Coulomb [3] improved this result by proving that for z0 purely imaginary and  not real, the real part of  cannot be greater than − 32 . In the same paper he proved that there exists a ∗

Corresponding author. E-mail address: [email protected] (E.N. Petropoulou). 1 Supported by the State Scholarship’s Foundation.

c 2002 Elsevier Science B.V. All rights reserved. 0377-0427/03/$ - see front matter  PII: S 0 3 7 7 - 0 4 2 7 ( 0 2 ) 0 0 6 4 1 - 6

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sequence of real numbers i i = 1; 2; : : : ; −∞ ¡ · · · ¡ 2 ¡ 1 ¡ + ∞ such that Ji (z0 ) = 0. For this result, alternative proofs were given in [11] and [9, p. 465]. Also in [6], in the study of a class of combinatorial problems called non-overlapping partitions, Flajolet and Schott showed that equation J (2) = 0 has no positive zeros , i.e., there do not exist positive , so as 2 to be a common zero of J (z). Regarding numerical results in this direction, we should mention that Benton and Knoble [2] showed that it is possible for two Bessel functions J (z) and J (z) to have two positive zeros z1 ; z2 in common. More precisely, they developed a computer program which takes rough approximations of the orders ;  and the position of the common zeros and derives the exact values. Later, in [1], Benton developed a method for obtaining the initial approximations of the orders ;  and the common zeros z1 ; z2 . Finally we should also mention that Ikebe et al. [10] showed that the simple zeros of J (z0 ), with respect to , can be approximated by the eigenvalues of complex symmetric tridiagonal matrices m × m; m → ∞ and provided an estimation for the error. Our results in this direction are Theorems 3.2–3.4 and Corollary 3.1. The other way of dealing with the problem of common zeros of Bessel functions is to consider 5xed i ; i = 1; 2; : : : ; k; k ¿ 2, and to locate those  which are common zeros of Ji (z). Our contribution in this direction is Theorem 3.5. In Section 2, we give some de5nitions, notations and results that we are going to need in our functional-analytic approach. Finally in Section 3, we present our main results. 2. Preliminaries Denote by H an abstract separable Hilbert space over the complex 5eld, by {en }; n = 1; 2; : : : its orthogonal base and by (·; ·) and · the usual inner product and norm, respectively, in H . In our study we will need the following operators. The shift operator V de5ned by Ven = en+1 ; and its adjoint

V∗

n = 1; 2; : : : de5ned by

V ∗ en = e n− 1 ;

n = 2; 3; : : : ;

V ∗ e1 = 0:

The self-adjoint operator T0 = V + V ∗ for which the spectrum is purely continuous and covers the entire interval [ − 2; 2] [7]. Also T0 is bounded with T0 = 2 and (T0 f; f) 6 2;

f = 1:

(2.1)

The self-adjoint diagonal operator C0 de5ned by C0 en = nen ;

n = 1; 2; : : :

has a self-adjoint extension with discrete spectrum [8], i.e., the de5nition domain of C0 can be extended to the range domain of the bounded operator B0 de5ned by 1 B0 en = en ; n = 1; 2; : : : : n

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389

For C0 the following inequality is easily found to hold: (C0 f; f) ¿ 1;

f = 1:

(2.2)

Also we observe that the operator −C0 is the in5nitesimal generator of the following strongly continuous semigroup: T (t)x =

∞


e−nt (x; en )en ;

x ∈ H; t ¿ 0;

(2.3)

n=1

since

   T (t)x − x    → 0: + C x 0   t →0 t

Moreover the semigroup de5ned by (2.3) is a contraction semigroup, since T (t) 6 e−t 6 1;

for t ¿ 0:

(2.4)

The diagonal operator L de5ned by L en =

1 en ; +n

n = 1; 2; : : : ;  = −n;

which is self-adjoint for  ∈ R. The operator L is the compact inverse of the operator C0 + I and for Re  ¿ − 1 has the following integral representation [15]:  ∞ L x = e−t T (t)x dt; x ∈ H (2.5) 0

in the sense that (L x; f) =

 0

∞

e−t (T (t)x; f) dt;

∀f ∈ H;

where T (t) is de5ned by (2.3). 3. Main results First of all we recall the following basic theorem which was proved in [13,9]: Theorem 3.1. A number  = 0;  ∈ C is a zero of the Bessel function J (z) if and only if 2= is an eigenvalue of the operator L T0 . The main results of the present paper are the following: Theorem 3.2. Let  = 1 + i2 = 0; 1 ; 2 ∈ R, be a complex number. Then  is a common zero of the Bessel functions Jm (z), m = 1; 2; : : : ; k; k ¿ 2, m = m1 + im2 ; m = −1; −2; : : : ; if and only if the numbers (2=)m , m = 1; 2; : : : ; k are eigenvalues of the operator T0 − (2=)C0 . In this case

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the following inequalities hold: m1 6 |1 | − 1;

(3.1)

|m2 | 6 |2 |:

(3.2)

Proof. The proof of the 5rst part of the theorem follows easily from Theorem 3.1 and the following relationships:   2 2 2 2m Lm T0 xm = xm ⇔ T0 xm = (C0 + m )xm ⇔ T0 − C0 xm = (3.3) xm ;     where xm = xm (m ) = 0, xm = 1. Taking the inner product of both parts of (3.3) with xm and equating the real and imaginary parts of the resulting equation we obtain 1 m 1 = (T0 xm ; xm ) − (C0 xm ; xm ); 2 2 (T0 xm ; xm ): m 2 = 2 By taking into consideration inequalities (2.1) and (2.2) we obtain m1 6 |1 | − 1; |m2 | 6 |2 |: Remark 3.1. The hypothesis m = −n; n = 1; 2; : : : is necessary for the operators Lm to be de5ned on all H . However, this is not a restriction for studying the zeros of J−n (z), since those coincide with the zeros of Jn (z), due to the relationship J−n (z) = (−1)n Jn (z); n = 1; 2; : : : : Remark 3.2. If (3.1) or (3.2) do not hold, then  cannot be a common zero of Jm (z); m = 1; 2; : : : ; k; k ¿ 2. In other words, if  is a common zero of Jm (z); m = 1; 2; : : : ; k; k ¿ 2, then m must lie in the strip: {m = m1 + im2 ∈ C: m1 6 |Re | − 1 and |m2 | 6 |Im |}: Remark 3.3. In case  = i2 is a purely imaginary common zero of Jm (z), m = 1; 2; : : : ; k, k ¿ 2, we have that m 1 6 − 1

and

|m2 | 6 |2 |:

The above result improves the result of Dougall [4] and Coulomb [3], concerning the imaginary part of m , and con5rms the numerical results of [10] obtained for  = 6i. Theorem 3.3. Given a complex number  = 0, with || ¡ 12 , there exists a sequence of complex numbers m ∈ Dm = { ∈ C: | + m| 6 ||}, such that Jm () = 0, m = 1; 2; : : : :

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Proof. According to Theorem 3.2,  = 1 + i2 = 0 is a zero of Jm (z) if and only if, −m are eigenvalues of the operator C0 − (=2)T0 , i.e.,    C0 − T0 xm = −m xm ; xm = 1: 2 The operator C0 is self-adjoint with point spectrum {1; 2; 3; : : :} and has a positive de5nite and Hilbert–Schmidt inverse. The operator (=2)T0 ,  ∈ C, is not self-adjoint with − (=2)T0 = || ¡ 12 ¡ 1 = 1 , where 1 is the 5rst eigenvalue of C0 . Then according to a result of Osborn [12, Theorem 2.2, p. 395], there exists a sequence of eigenvalues of C0 − (=2)T0 , each of which is contained in one of the disks Dm , m = 1; 2; : : : : Remark 3.4. As it was mentioned in the introduction, it was proved in [9, p. 464] that when  is real, there exists a sequence of real numbers |m | → ∞, m → ∞, such that Jm () = 0, m = 1; 2; : : : : Theorem 3.4. The number of di?erent Bessel functions Jm (z), m = 1; 2; : : : ; k, k ¿ 2, m ¿ 0, which have a common zero  ¿ 0 is equal to the number of negative eigenvalues of the operator C0 − (=2)T0 . Proof. The proof follows from Theorem 3.2 taking into consideration Remark 3.4 and the relation 2 2   C0 − T0 : T0 − C0 = −   2 Corollary 3.1. A su9cient condition for the set of negative eigenvalues of the operator C0 −(=2)T0 to be empty is 0 ¡  6 1. Proof. Assume that 0 ¡  6 1 and that w is an eigenvalue of the operator C0 − (=2)T0 . Then    C0 − T0 x = wx; x = 1 2 from where it follows:  (C0 x; x) − (T0 x; x) = w: 2 Due to inequalities (2.1) and (2.2), we obtain from the previous relation, w ¿ 0, which proves the corollary. Remark 3.5. If 0 ¡  ¡ 1 then there do not exist i ¿ 0 such that Ji () = 0. Theorem 3.5. Let J (x) and J (x) be two Bessel functions of the :rst kind and orders  ¿  ¿ − 1. If  = 0 is a common zero of J (x) and J (x), the following lower bound for , || ¿ holds.

+1 ; 22 F1 (1= + 32 ; 1; 1 + 1=; −1)

=

2( − ) ; +1

(3.4)

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Proof. Since  is a common zero of J (x) and J (x), it follows from Theorem 3.1 that  = 2= is an eigenvalue of L T0 and L T0 , i.e., L T0 x = x;

x = 1;

(3.5)

L T0 y = y;

y = 1

(3.6)

or by using (2.5):  ∞ x = e−t T (t)T0 x dt; 0

 y =

0

∞

e−t T (t)T0 y dt

from where it follows:  ∞ T (t)T0 (e−t x + e−t y) dt (x + y) = 0



⇒ |x + y| =

∞

0

T (t) T0 e−t x + e−t y dt



⇒  x + y 6 2

0

∞

e−t e−t x + e−t y dt:

(3.7)

From the fact that  is a common zero of J (x) and J (x) it follows, by use of Theorem 3.2, that the normalized eigenelements x and y of L T0 and L T0 , respectively, are also eigenelements of the self-adjoint operator T0 − (2=)C0 , corresponding to the eigenvalues 2=, 2=, respectively. Set k =  − . Since  = , the eigenelements x and y are orthogonal. Bearing in mind this fact, relation (3.7), and applying the following change of variables e−t = u1=(+1) ,  = 2k=( + 1) = 2( − )=( + 1) ¿ 0, we obtain √  ∞ −t  e e−2t + e−2t dt || 6 2 √

0

2 = ( + 1)

 0

1

u

1=−1

√

4 1 + u du = 2 F1 +1



 1 1 3 + ; 1; 1 + ; −1 :  2 

Here we used the identity 2 F1 (; b; c; z) = (1 − z)2c−−b F1 (c − ; c − b; c; z) (see (1) and (2) [5, p. 105]). Remark 3.7. For ; ;  ∈ R,  = −n;  = −n; n = 1; 2; : : : it is easy to see from (3.5) and (3.6) that (x + y) = L T0 x + L T0 y;

x = y = 1;

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from where it follows: || x + y 6 L · T0 · x + L · T0 · y √ ⇒ || 2 6 √

2 2 + | + 1| | + 1|

2| + 1| · | + 1| : (3.10) | + 1| + | + 1| For ;  ¿ − 1 and  ∈ R as in Theorem 3.4, the lower bound for  obtained by (3.4) is better than the corresponding lower bound obtained by (3.10). ⇒ || ¿

Remark 3.8. If  does not satisfy (3.4) or (3.10), then it cannot be a common zero of J (z), J (z). Acknowledgements The authors would like to thank the referee for his/her useful suggestions which helped to improve the presentation of this paper. References [1] T.C. Benton, Common zeros of two Bessel functions Part II. Approximation and tables, Math. Comp. 41 (163) (1983) 203–217. [2] T.C. Benton, H.D. Knoble, Common zeros of two Bessel functions, Math. Comp. 33 (142) (1978) 533–535. [3] M.J. Coulomb, Sur les zMeros de fonctions de Bessel considMerMees comme fonctions de l’ ordre, Bull. Sci. Math. 60 (1936) 297–302. [4] J. Dougal, The determination of Green’s function by means of cylindrical and spherical harmonics, Proc. Edinburgh Math. Soc. 18 (1900) 33–83. [5] A. ErdMelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953. [6] P. Flajolet, R. Schott, Non-overlapping partitions, continued fractions, Bessel functions and a divergent series, European J. Combin. 11 (1990) 421–432. [7] E.K. Ifantis, On the nature of the spectrum of generalized oscillator phase operators, Lett. Nuovo Cimento 2 (21) (1971) 1096–1100. [8] E.K. Ifantis, An existence theory for functional-di?erential equations and functional-di?erential systems, J. Di?erential Equations 29 (1) (1978) 86–104. [9] E.K. Ifantis, P.D. Siafarikas, C.B. Kouris, Conditions for solutions of a linear 5rst-order di?erential equation in the Hardy–Lebesque space and applications, J. Math. Anal. Appl. 104 (2) (1984) 454–466. [10] Y. Ikebe, N. Asai, Y Miyazaki, D. Cai, The eigenvalue problem for in5nite complex symmetric tridiagonal matrices with application, Linear Algebra Appl. 241–243 (1996) 599–618. [11] D. Maki, On constructing distribution functions with applications to Lommel polynomials and Bessel functions, Trans. Amer. Math. Soc. 103 (2) (1968) 281–297. [12] J.E. Osborn, Approximation of the eigenvalues of non self-adjoint operators, J. Math. Phys. 45 (1966) 391–401. [13] P.D. Siafarikas, Analytic solutions of singular linear di?erential equations and applications, Ph.D. Thesis, Patras, Greece, 1980 (in Greek). [14] G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Edition, Cambridge University Press, Cambridge, 1944. [15] K. Yosida, Functional Analysis, Springer, Berlin, 1965.