Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 57956, 7 pages
doi:10.1155/2007/57956
Research Article
Product Bessel Distributions of the First and Second Kinds
Saralees Nadarajah
Received 15 August 2005; Accepted 11 February 2007
Recommended by Andrew Rosalsky
A new Bessel function distribution is introduced by taking the product of a Bessel function pdf of the first kind and a Bessel function pdf of the second kind. Various particular
cases and expressions for moments are derived.
Copyright © 2007 Saralees Nadarajah. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Univariate Bessel function distributions have been used to model signal output processed
by a radar receiver under various sets of conditions (see, e.g., McNolty [1]). There are two
kinds of univariate Bessel function distributions. Bessel function distribution of the first
kind has the pdf given by
1 − c2 m+1/2 xm
cx
x
Im
exp −
f (x) = √ m m+1
π2 b Γ(m + 1/2)
b
b
(1.1)
for x > 0, b > 0, c > 1 and m > 1, where
xm
Im (x) = √ m
π2 Γ(m + 1/2)
1
−1
1 − t2
m−1/2
exp(±xt)dt
(1.2)
is the modified Bessel function of the first kind. Bessel function distribution of the second
kind has the pdf given by
1 − c2 m+1/2 |x|m
x
cx
exp −
Km f (x) = √ m m+1
b
π2 b Γ(m + 1/2)
b
(1.3)
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International Journal of Mathematics and Mathematical Sciences
for −∞ < x < ∞, b > 0, |c| < 1, and m > 1, where
√
πxm
Km (x) = m
2 Γ(m + 1/2)
∞
t2 − 1
1
m−1/2
exp(−xt)dt
(1.4)
is the modified Bessel function of the second kind. In thispaper, we introduce a new
Bessel function distribution with its pdf taken to be the product of two densities of the
form (1.1) and (1.3), that is,
x
x
Kn
b
β
f (x) = Cxm+n Im
(1.5)
for x > 0, 0 < β < b, m > 1, and n > 1, where C denotes the normalizing constant. Application of [2, equation (2.16.28.1)] by Prudnikov et al. shows that one can determine C
as
1
1
1 2m+n−1 β2m+n+1
=
Γ m+n+ Γ m+
m
C
b Γ(m + 1)
2
2
2 F1
β2
1
1
m + n + ,m + ;m + 1; 2 ,
2
2
b
(1.6)
where 2 F1 is the Gauss hypergeometric function defined by
2 F1 (a,b;c;x) =
∞
(a)k (b)k xk
k =0
(c)k
,
k!
(1.7)
where ( f )k = f ( f + 1) · · · ( f + k − 1) denotes the ascending factorial. Using special properties of the Gauss hypergeometric function, one can obtain simpler expressions for (1.6).
For instance, if m = n, then (1.6) can be reduced to
−m 2m + 1
1
= π −1/2 22m−1 (bβ)2m+1/2 b2 − β2
Γ
C
2
2
2
b
+
β
m
,
× exp(miπ)Qm
−1/2
2bβ
(1.8)
μ
where Qν (·) is the Legendre function defined by
μ
Qν (x) =
√
μ/2
π exp(iμπ)Γ(μ + ν + 1) −μ−ν−1 2
μ+ν+1 μ+ν
3 1
x − 1 2 F1
x
,
;ν + ; 2 .
2ν+1 Γ(ν + 3/2)
2
2
2 x
(1.9)
In the rest of thispaper, we derive various expressions for particular forms of (1.5) and its
moments.
Saralees Nadarajah 3
2. Particular cases
When m and n take half-integer values, one can reduce (1.5) to elementary forms. Note
that
I3/2 (x) =
2 x cosh(x) − sinh(x)
,
π
x3/2
I5/2 (x) =
2 x2 + 3 sinh(x) − 3x cosh(x)
,
π
x5/2
I7/2 (x) =
2 x x2 + 15 cosh(x) − 3 2x2 + 5 sinh(x)
,
π
x7/2
I9/2 (x) =
2 x4 + 45x2 + 105 sinh(x) − 5x 2x2 + 21 cosh(x)
π
x9/2
(2.1)
and, more generally, if ν − 1/2 ≥ 1 is an integer, then
√ √
Iν (x) = 2 xπ exp
πi 1
−ν
2 2
|ν| + 2k − 1/2 !
(2k)! |ν| − 2k − 1/2 !(2x)2k
k =0
[(2|ν
|−3)/4] |ν| + 2k + 1/2 !(2x)−2k−1
πx 1
.
+ cosh
−ν −x
2 2
(2k + 1)! |ν| − 2k − 3/2 !
k =0
πx 1
× sinh
−ν −x ×
2 2
[(2|ν
|−1)/4]
(2.2)
Furthermore, note that
π exp(−x)(x + 1)
,
2
x3/2
π exp(−x) x2 + 3x + 3
K5/2 (x) =
,
2
x5/2
π exp(−x) x3 + 6x2 + 15x + 15
K7/2 (x) =
,
2
x7/2
4
π exp(−x) x + 10x3 + 45x2 + 105x + 105
K9/2 (x) =
2
x9/2
K3/2 (x) =
(2.3)
and, more generally, if ν − 1/2 ≥ 1 is an integer, then
√
√
Iν (x) = π exp(−x) 2x
[|ν|−
1/2]
j =0
j + |ν| − 1/2 !(2x)− j
.
j! |ν| − j − 1/2 !
(2.4)
Thus, several particular forms of (1.5) can be obtained for half-integer values of m and n.
For example, if m = 3/2 and n = 3/2, then (1.5) reduces to
f (x) = C(bβ)3/2
x
x
x
cosh
− sinh
b
b
b
exp −
x
β
x
+1 .
β
(2.5)
International Journal of Mathematics and Mathematical Sciences
0.3
0.2
0.2
pdf
0.3
pdf
4
0.1
0.1
0
0
0
1
2
3
4
5
0
1
2
x
3
4
5
3
4
5
x
(a)
(b)
0.2
0.2
pdf
0.3
pdf
0.3
0.1
0.1
0
0
0
1
2
3
4
5
0
1
2
x
x
(c)
(d)
Figure 2.1. Plots of the pdf (1.5) for b = 1, β = 1/2, and (a) m = 1.1; (b) m = 1.3; (c) m = 1.5; and,
(d) m = 2. The four curves in each plot from the left to the right correspond to n = 1.1,1.3,1.5,2.
If m = 3/2 and n = 5/2, then (1.5) reduces to
f (x) = Cb3/2 β5/2
x
x
x
− sinh
cosh
b
b
b
exp −
x
β
x2 3x
+
+3 .
β2 β
(2.6)
Figure 2.1 illustrates possible shapes of the pdf (1.5) for selected values of m and n.
The four curves in each plot correspond to selected values of n. Note that the shapes are
unimodal and that the densities appear to shrink with increasing values of both m and n.
Saralees Nadarajah 5
3. Moments
If X is a random variable with pdf (1.5), then its kth moment can be expressed as
E X
k
=C
∞
0
x
x
Kn
dx.
b
β
xk+m+n Im
(3.1)
Application of [2, equation (2.16.28.1)] by Prudnikov et al. shows that (3.1) can be calculated as
E Xk =
C2k+m+n−1 βk+2m+n+1
k+1
k+1
Γ m+
Γ m+n+
bm Γ(m + 1)
2
2
2
β
k+1
k+1
× 2 F1 m + n +
,m +
;m + 1; 2 .
2
2
b
(3.2)
Using special properties of the Gauss hypergeometric function, one can derive several
simpler forms of (3.2) as discussed in the following. If m = n, then (3.2) reduces to
E X k = Cπ −1/2 2k+2m−1 (bβ)k+2m+1/2 b2 − β2
−(k+2m)/2
Γ
k + 2m + 1
2
(k + 2m)iπ (k+2m)/2 b2 + β2
Qm−1/2
.
× exp
2
2bβ
(3.3)
If k ≥ 1 is odd, then (3.2) can be reduced to the following elementary form:
C2k+m+n−1 bk+m+2n+1 βk+2m+n+1
k+1
k+1
Γ m+n+
Γ m+
E Xk = m+n+(k+1)/2
2
2
2
2
b −β
Γ(m + 1)
β2
k+1 1−k
× 2 F1 m + n +
,
;m + 1; 2
2
2
β − b2
C2k+m+n−1 bk+m+2n+1 βk+2m+n+1
k+1
k+1
Γ m+n+
Γ m+
= m+n+(k+1)/2
2
2
b2 − β2
Γ(m + 1)
×
(k−
1)/2
j =0
m + n + (k + 1)/2
(1 − k)/2 j j
β2
β2 − b2
(m + 1) j
(3.4)
j
.
When k is even, one can reduce (3.2) to simpler forms when m and n take integer or
half-integer values. If either both m and n are half-integers or m is an integer and n is
a half-integer or m is a half-integer and n is an integer, then (3.2) can be reduced to an
elementary form. On the other hand, if both m and n are integers, then one can express
(3.2) in terms of the complete elliptical integral of the first kind and the complete elliptical
integral of the second kind defined by
EllipticK(a) =
1
0
√
dx
√
dx,
1 − a2 x 2
1 − x2
1 √
1 − a2 x 2
√
dx,
EllipticE(a) =
1 − x2
0
(3.5)
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International Journal of Mathematics and Mathematical Sciences
respectively. For instance, if m = 3/2 and n = 3/2, then the first four even order moments
are
−35 − 14x + x2
,
b3/2 (−1 + x)5
−105 − 189x − 27x2 + x3
E X 4 = 144Cβ19/2
,
b3/2 (−1 + x)7
−231 − 924x − 594x2 − 44x3 + x4
E X 6 = 5760Cβ23/2
,
b3/2 (−1 + x)9
−429 − 3003x − 4290x2 − 1430x3 − 65x4 + x5
E X 8 = 403200Cβ27/2
,
b−3/2 (−1 + x)−11
E X 2 = 8Cβ15/2
(3.6)
where x = β2 /b2 and the normalizing constant C = 2β11/2 (−5 + x)/ {b3/2 (−1 + x)3 }. If m =
2 and n = 2, then the first four even order moments are
√
√
√
E X 2 = 15Cβ9 − 23EllipticK( x)x − 87EllipticK( x)x2 + 107EllipticK( x)x3
√
√
√
+ EllipticK( x)x4 + 2EllipticK( x) + 22EllipticE( x)x
√
√
√
+ 216EllipticE( x)x2 + 22EllipticE( x)x3 − 2EllipticE( x)x4
√ − 2EllipticE( x) / x2 b2 (−1 + x)6 ,
√
√
√
E X 4 = 315Cβ11 − 39EllipticK( x)x − 536EllipticK( x)x2 + 158EllipticK( x)x3
√
√
√
4
5
+ 414EllipticK( x)x + EllipticK( x)x + 2EllipticK( x)
√
√
√
+ 38EllipticE( x)x + 988EllipticE( x)x2 + 988EllipticE( x)x3
√
√
+ 38EllipticE( x)x4 − 2EllipticE( x)x5
√ − 2EllipticE( x) / x2 b2 (−1 + x)8 ,
√
√
√
E X 6 = 2835Cβ13 − 295EllipticK( x)x − 8771EllipticK( x)x2 − 8886EllipticK( x)x3
√
√
√
4
5
6
+ 12452EllipticK( x)x + 5485EllipticK( x)x + 5EllipticK( x)x
√
√
√
+ 10EllipticK( x) + 290EllipticE( x)x + 14546EllipticE( x)x2
√
√
+ 35884EllipticE( x)x3 + 290EllipticE( x)x5
√
√
+ 14546EllipticE( x)x4 − 10EllipticE( x)x6
√ − 10EllipticE( x) / (−1 + x)10 x2 b2 ,
√
√
√
E X 8 = 155925Cβ15 14EllipticK( x) − 581EllipticK( x)x − 30336EllipticK( x)x2
√
√
− 86111EllipticK( x)x3 + 19958EllipticK( x)x4
√
√
5
6
+ 80445EllipticK( x)x + 16604EllipticK( x)x
√
√
√
+ 7EllipticK( x)x7 − 14EllipticE( x) + 574EllipticE( x)x
√
√
√
√
+ 47514EllipticE( x)x2 + 214070EllipticE( x)x3
+ 47514EllipticE( x)x5 + 214070EllipticE( x)x4
√
√
− 14EllipticE( x)x7 + 574EllipticE( x)x6 / (−1 + x)12 x2 b2 ,
(3.7)
Saralees Nadarajah 7
where x = β2 /b2 and the normalizing constant C satisfies
√
√
√
√
1
= 3β7 − 11EllipticK( x)x + 8EllipticK( x)x2 + EllipticK( x)x3 + 2EllipticK( x)
C
√
√
√
+ 10EllipticE( x)x + 10EllipticE( x)x2 − 2EllipticE( x)x3
√ − 2EllipticE( x) / x2 b2 (−1 + x)4 .
(3.8)
References
[1] F. McNolty, “Applications of Bessel function distributions,” Sankhyā, vol. 29, pp. 235–248, 1967.
[2] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 2, Gordon & Breach
Science Publishers, New York, NY, USA, 1986.
Saralees Nadarajah: School of Mathematics, University of Manchester, Manchester M13 9PL, UK
Email address: saralees.nadarajah@manchester.ac.uk