arXiv:1105.0957v1 [math-ph] 4 May 2011
Approximate closed-form formulas for the
zeros of the Bessel Polynomials
Rafael G. Campos and Marisol L. Calderón
Facultad de Ciencias Fı́sico-Matemáticas,
Universidad Michoacana,
58060, Morelia, Mich., México.
[email protected], [email protected]
MSC: 33C47, 33F05, 33C10, 30C15
Keywords: Orthogonal polynomials, Bessel polynomials, Zeros, Asymptotic
expressions, Rate of convergence.
Abstract
We find approximate expressions x̃(k, n) and ỹ(k, n) for the real and imaginary parts of the kth zero zk = xk + iyk of the Bessel polynomial yn (x).
To obtain these closed-form formulas we use the fact that the points of welldefined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically
through an implementation of the electrostatic interpretation formulas and
then, a fit to the real and imaginary parts as functions of k and n is obtained. It is shown that the resulting complex number x̃(k, n) + iỹ(k, n) is
O(1/n2)-convergent to zk for fixed k.
1
1
Introduction
The polynomial solutions of the differential equation
x2 y ′′(x) + 2(x + 1)y ′(x) − n(n + 1)y(x) = 0
(1)
were studied systematically in [1] by the first time. They are named Bessel
polynomials and are given explicitly by
n
X
(n + k)! x k
yn (x) =
,
(n − k)!k! 2
(2)
k=0
where n = 0, 1, · · · . Many properties as well as applications are associated
to these polynomials: the solution of the wave equation in spherical coordinates, network and filter design, isotropic turbulence fields, and more (see
the monograph [2] or [3]-[14] and references therein for some other results).
Among these, several results about the important problem concerning the
location of its zeros have been obtained [8]-[11]. Just for instance, the use of
the reverse Bessel polynomials θn (x) = xn yn (1/x) in filter design is known
since the time when these polynomials began to be studied [2, 14]. Here, the
poles of the transfer function are essentially the zeros of θn (x). Thus, it is
desirable to acquire new analytical knowledge about the location of the zeros
of the Bessel polynomials.
In this note we give approximate explicit formulas for both the real and
imaginary parts of the kth zero zk = xk + iyk of yn (x) and show that the
approximation order of these new formulas to the exact zeros of the Bessel
polynomials is O(1/n2 ) for fixed k.
The approach followed in this note is simple and based on three items. The
first is the electrostatic interpretation of the zeros of polynomials satisfying
second order differential equations [15]-[17], the second is a simple curve fitting of numerical data and the third is the known fact that the points of
well-defined curves in the complex plane are limit points of the zeros of the
normalized Bessel polynomials [8]-[11]. The formulas yielded by the electrostatic interpretation of the zeros of Bessel polynomials are used to find
them numerically as it has been done previously with these and other sets of
points [7]-[19]. Several sets of zeros are computed in this way and the sets of
real and imaginary values are fitted by polynomials depending on the index
k whose coefficients depend on n. Finally, it is found that the approximate
expression for the kth zero of yn (x) is O(1/n2 )-convergent to a limit point
2
of the zeros of the Bessel polynomials. Since the exact zero zk is O(1/n2)convergent to its limit point [8], we conclude that the approximation order
of our approximate expression to the exact kth zero is also O(1/n2).
2
Asymptotic expressions for the zeros
Let zk = xk + iyk , k = 1, 2, · · · , n, be the zeros of the Bessel polynomial
yn (x). Then, from (1) follows that
N
X
k=1
1
(zj + 1)
+
= 0,
zj − zk
zj2
where j = 1, 2, · · · , n, i.e., the real and imaginary parts of the zeros should
satisfy the electrostatic equations
N
X
x3j + x2j + xj yj2 − yj2
xj − xk
+
= 0,
2
(xj − xk )2 + (yj − yk )2
x2j + yj2
k=1
N
X
yj x2j + 2xj + yj2
yj − yk
+
= 0.
2
(xj − xk )2 + (yj − yk )2
x2j + yj2
k=1
(3)
This set of nonlinear equations can be solved by standard methods. We have
used a Newton method to solve them up to n = 500. As it is shown in Fig. 1,
the piecewise linear interpolation of the real and imaginary parts of the zeros
of the normalized Bessel polynomials yn (x/n) can be fitted by polynomials
of the second and third degree in the index k.
0.0
1.0
-0.2
0.5
-0.6
ImHzL
ReHzL
-0.4
-0.8
0.0
-1.0
-0.5
-1.2
-1.4
-1.0
0
100
200
300
400
500
0
k
100
200
300
400
500
k
Figure 1: Real and imaginary parts of the zeros of the normalized Bessel
polynomials yn (x/n) for n = 100, 200, 300, 400, and 500, plotted in gray-level
intensity, from lower to higher, according to the value of n.
3
Thus, we propose the following expressions
x̃(k, n) = a2 (n)k 2 + a1 (n)k + a0 (n),
ỹ(k, n) = b3 (n)k 3 + b2 (n)k 2 + b1 (n)k + b0 (n)
(4)
to fit our data. To find the relationship between the coefficients of these
polynomials and n, we take into account the numerical behavior of the data
at the middle and end points. According to this, x̃(k, n) and ỹ(k, n) can be
determined by
n
3
x̃( , n) = − ,
2
2n
x̃(0, n) = 0,
x̃(n + 1) = 0,
(5)
and
1
ỹ(1, n) = − ,
n
n+1
ỹ(
, n) = 0,
2
dỹ(k, n) 4
n = 2,
k= 2
dk
n
ỹ(n, n) =
1
. (6)
n
These conditions lead to the following coefficients
6(n + 1)
6
,
a
(n)
=
−
,
a0 (n) = 0,
1
n2 (n + 2)
n2 (n + 2)
12(n − 2)(n + 1)
8(n − 2)
,
b
(n)
=
, (7)
b3 (n) = − 2 3
2
n (n − 3n2 + 2)
(n − 1)n2 (n2 − 2n − 2)
2 (n3 + 6n2 − 12n − 4)
n3 − 5n2 + 6
b1 (n) = −
,
b
(n)
=
−
.
0
(n − 1)n2 (n2 − 2n − 2)
n (n3 − 3n2 + 2)
a2 (n) =
The substitution of (7) in (4) yields approximate closed-form expressions
z̃k = x̃(k, n) + iỹ(k, n),
(8)
k = 1, 2, · · · , n, that converge to the zeros zk of the Bessel polynomial yn (x),
as we will show in the following.
3
Convergence
Following [9], we define
√
2
e 1+1/z
p
W (z) =
z(1 + 1 + 1/z 2 )
4
(9)
and denote by Γ the curve defined by
Γ = {z ∈ C : |W (z)| = 1 and |argz| ≥
π
},
2
which contains the limit points ω̂k of the zeros of the normalized Bessel
polynomial yn (x/n). Then, it has been proved in [8] that the zero ωk of
yn (x/n) approaches to order O(1/n) the limit value ω̂k , i.e,
|ωk − ω̂k | = O(1/n),
(10)
as n → ∞.
Thus, if we show that |ω̃k − ω̂k | = O(1/n), we will have proved that
|ωk − ω̃k | = O(1/n),
(11)
and therefore, taking into account that ωk = nzk , the explicit expression (8)
approaches to order O(1/n2) the zero zk of the Bessel polynomial yn (x).
To this purpose, we simply substitute ω̃k = nz̃k in (9) to obtain, after a
lengthily calculation, that the expansion of W (ω̃k ) in terms of 1/n is
√
6k
2
2
W (ω̃k ) = 1 − 2 10k − 2k + 1 sin
4k − 4
√
1
1
6k
2
2
10k − 2k + 1 cos
−
+k−1
,
+O
4k − 4
n
n3/2
and this implies that
1
|W (ω̃k )| = 1 + O
n
for fixed k. Thus, ω̃k approaches to order O(1/n) the Γ curve and (11)
follows. From here we have that
|zk − z̃k | = O(1/n2)
(12)
as n → ∞. Numerical calculations confirm and extend this result. Figure 2
shows the behavior of the maxima of |zk − z̃k | over k as they depend on n.
The numbers computed by (3) are taken as the exact zeros zk . The displayed
data shows numerical uniform convergence on the values of k, not only for
fixed k. A fit of these data gives 1/na with a = 1.7.
5
0.0010
~
max zk -z k 0.0008
k
0.0006
0.0004
0.0002
100
200
300
n
400
500
Figure 2: Plot of the values of maxnk=1 |zk − z̃k | against n.
4
Some few tests
Just to give examples of the application of the approximate expression (8),
we consider the following cases.
4.1
The real zero
An closed-form formula for the unique real zero α(n) of the Bessel polynomial
yn (x) can be obtained by the substitution of k = (n + 1)/2 in the real part
of (8), x̃(k, n). This gives
α(n) = −
3(n + 1)2
+ O(1/n2).
2n2 (n + 2)
(13)
as our new result. In [2, 11] asymptotic expressions for α(n) are given.
Particularly, the following formula
α(n) = −
2
+ O(1/n),
1.32549n + 0.662743
(14)
can be obtained from a more general expression given in [11]. Deleting the
O-terms and expanding both results in powers of 1/n we find that
α̃(n) ≃ −
3
,
2n
1
α(n) ≃ −1.50888 ,
n
indicating good agreement between the two approaches.
6
4.2
Power sums
Here we carry out the corresponding multiplications and use some cases of
Faulhaber’s formula. Then we compare our results with the exact ones.
1. Sum of the zeros. The simple sum of z̃k [cf. (8)], gives
s̃1 (n) =
n
X
k=1
z̃k = −
n+1
= −1 + O(1/n).
n
The exact result is s1 (n) = −1, as can be seen from (2).
2. Sum of the squares of the zeros. In this case we obtain
s̃2 (n) =
n
X
z̃k2 =
k=1
p2 (n)
q2 (n)
where
p2 (n) = 55n8 + 15n7 − 800n6 − 612n5 + 4064n4 + 1740n3
− 1696n2 − 2832n − 3312,
2
q2 (n) = 105n3 n2 − 2n − 2 n2 + n − 2 .
The exact sum
1
,
2n − 1
can be found elsewhere [12]. Expanding both expressions in powers of 1/n
we find that
11
1
s̃2 (n) =
+ O(1/n2),
s2 (n) =
+ O(1/n2).
21n
2n
3. Sum of the cubes of the zeros. In a similar form, we obtain that
s2 (n) =
s̃3 (n) =
n
X
k=1
z̃k3 =
p3 (n)
q3 (n)
where
p3 (n) = −n10 + 28n9 − 217n8 + 468n7 + 1002n6 − 3804n5 − 1076n4 + 5936n3
− 3848n2 + 12816n + 19584,
2
q3 (n) = 35(n − 1)n5 n3 − 6n − 4 .
In this case, the exact sum is zero [12]. Expanding s̃3 (n) in 1/n we find that
s̃3 (n) = O(1/n2 ).
7
5
Final comment
Note that the approximate formula for zk given above is not unique. There
exist other functions to fit the zeros obtained through the electrostatic equations (3), and there are other conditions to impose at the extreme and middle
points of the fitting interval. For instance, the imaginary part ỹ(k, n), can
be fitted by a polynomial of degree 5, but this does not improve the rate of
convergence and, on the other hand, the calculations become more lengthy.
6
Acknowledgment
The authors thank Consejo Nacional de Ciencia y Tecnologı́a for the financial
support given to this project.
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