Proyecciones Journal of Mathematics
Vol. 29, No 3, pp. 165-182, December 2010.
Universidad Católica del Norte
Antofagasta - Chile
ON THE DISTRIBUTIONS OF THE
DENSITIES INVOLVING NON-ZERO ZEROS
OF BESSEL AND LEGENDRE FUNCTIONS
AND THEIR INFINITE DIVISIBILITY
HEMANT KUMAR
D. A-V. P. G. COLLEGE KANPUR, INDIA
M. A. PATHAN
UNIVERSITY OF BOTSWANA, BOTSWANA
and
R. C. SINGH CHANDEL
D. V. P. G. COLLEGE ORAI, INDIA
Received : September 2009. Accepted : January 2010
Abstract
In the present paper, we introduce the probability density functions
involving non-zero zeros of the Bessel and Legendre functions. Then,
we evaluate the distributions of the characteristic functions defined
by these probability density functions and again obtain their related
functions and polynomials. Finally, we prove the infinite divisibility
of these probability density functions.
2000 Mathematics Subject Classification : 33C20, 62E15, 60E05,
60E10.
Keywords and Phrases : Probability density functions, characteristic functions, non zero zeros of Bessel and Legendre functions,
infinite divisibility.
166
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
1. Introduction
Hochstadt [4] has derived the infinite product representation of the Bessel
function of order real v, such that
¡ z ¢v
∞
Y
2
(1.1) Jv (z) =
Γ (v + 1) k=1
Ã
z2
1− 2
zk
!
, z ∈ C, zk 6= 0, ∀ k = 1, 2, ...
as considering the sequence of integrals
(1.2)
In(1)
1
=
2πi
Z
Cn
q
F1 (ξ)
dξ, i = (−1), n = 1, 2, ...
ξ (ξ − z)
where, the sequence of circles {Cn } about the origin such that Cn includes
all non-zero real zeros, ±z1 , ..., ±zn , of Bessel function Jv (z), the point z
such that for fixed z 6= zn and not passing through any zeros, also,
0 (z)
, G (z) = z −v Jv (z) , ∀ z ∈ C (the set of complex numbers).
F1 (z) = GG(z)
Again, Kumar and Srivastava [6] have obtained the infinite product
representation of the Legendre function such that
Pµ (z) = z
µ−2
∞
Y
k=1
Ã
1 − z2
1−
1 − ςk2
!
, z 6= 0, z ∈ C
and ςk 6= 0, and −1 < ςk < 1, ∀k = 1, 2...
(1.3)
by considering the sequence of integrals
(1.4)
In(2) =
1
2πi
Z
Cn
q
F2 (t)
dt, i = (−1), n = 1, 2, ...
(t − z)
where, the sequence of circles Cn about the origin such that Cn includes
all non-zero and real zeros, ±ς1 , ..., ±ς( n, ) lie between −1 to +1 (i.e. −1 <
ςk < 1, ∀ k = 1, 2, ..., n) of Legendre polynomial of degree µ, the point z
twice such that for fixed z6= ςn and not passes through any zeros. Also, it
( z)
−µ P (z), z 6= 0 and z ∈ C.
has F2 (z) = H
µ
H(z) , H(z) = z
The concept of infinite divisibility and the decomposition of distributions arise in probability and statistics in relation to seeking families of
probability distributions that might be a natural choice in certain applications, in the same way that the normal distribution is. The term infinitely
On the distributions of the densities involving non-zero ...
167
divisible characteristic function is used for the characteristic function of
any infinitely divisible distribution.These distributions play a very important role in probability theory in the context of limit theorems. Takano
[10] has conjectured that following probability density in normed conjugate
product of Gamma functions
2ix
2
1
2
³
Γ (1 − ix) Γ (1 + ix) =
= Q
π
sin πix
π ∞ 1+
k=1
x2
k2
´ , −∞ < x < ∞
(1.5)
is infinite divisible.
Several authors studied infinite divisibility of many functions and their
related topics (See,, Bondesson [2], Goovaerts, D Hooge and Pril [3], Kelkar
[5], Steutel [8], Takano [9, 10] and Thorin [11] etc.). Motivated by the above
work and from (1.1), (1.3) and (1.5), we introduce following equalities in
the form of probability density functions:
(1.6)
and
(1.7)
1
(ix)v
µ
=
Q∞
2v Γ (v + 1) Jv (ix)
k=1 1 +
x2
zk2
¶ , −∞ < x < ∞
1
(ix)µ
µ
¶ , −∞ < x < ∞
=
2
Q
x2 +ςk2
x Pµ (ix)
∞
k=1
1−ςk2
respectively, and where, zk 6= 0 and ςk 6= 0 − 4, and −1 < ςk < 1,
∀ k = 1, 2..., n, are zeros of Bessel and Legendre functions, respectively.
Here, we conjecture that above densities (1.6) and (1.7) are infinitely divisible. Therefore, for that we define following probability density functions:
(1.8)
Qn
f1 (x) = λ1 Qn
k=1
¡
k=1
and
(1.9)
¡ 2¢
z
k
x2 + zk2
¢ , −∞ < x < ∞
¢
Qn ¡
2
k=1¡ 1 − ςk ¢
f2 (x) = λ2 Qn
2 , −∞ < x < ∞
2
k=1
x + ςk
168
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
where, zk 6= 0 and ςk 6= 0, and −1 < ςk < 1, ∀ k = 1, 2, ..., n, are zeros
of Bessel and Legendre functions, respectively and λ1 and λ2 are arbitrary
constants.
2. The Distributions of the Characteristic Function Involving
Non-zero Zeros of Bessel Function and its Related Polynomials
Theorem-1 : If in the complex upper half plane for the probability density
function (1.8) there exists a characteristic function
Z ∞
Φ1 (t) =
(2.1)
itx
e
Qn ¡ 2 ¢
zk
¢
Qn ¡k=1
2 dx, −∞ < t < ∞
2
λ1
x + zk
then, there holds the distribution formula of non-zero real zeros, zk ,
(i.e. zk 6= 0), ∀k = 1, 2, ..., n, of Bessel function on the real curve of t such
that
(2.2) Φ1 (t) = πλ1
−∞
n
X
k=1
zk
k=1
¡
Qn
¢
2
l=1,l6=k zl exp [−zk |t|]
¡ 2
¢
Qn
,
2
l=1,l6=k −zk + zl
−∞ < t < ∞
Proof : In the complex plane, consider the contour integral
Z
(2.3)
itz
e
Qn ¡ 2 ¢
zk
¢
Qn ¡k=1
2 dz, − − ∞ < t < ∞
2
λ1
z + zk
where, C is the closed contour consisting of γ, the upper half of the large
circle |z|
Q = R, the real axis from −R to R. The poles of the integrand
C
n
(z 2 )
k=1 k
2
(z +zk2 )
k=1
λ1
eitz Qn
k=1
are at z = ±izk , ∀k = 1, 2..., n and −∞ < t < ∞. Also,
inside C it has the simple
Q poles at z = izk , = 1, 2..., n in the upper half
n
(z 2 )
k=1 k
(z 2 +zk2 )
k=1
λ1
plane and lim|z|→∞ Qn
= 0 Hence by Cauchys residue theorem
and due to Jordan lemma, we find
Z ∞
itx
e
−∞
when t > 0.
(2.4)
Qn
(zk2 )
Qn k=1
2 dx
2
k=1 (x + zk )
λ1
= πλ1
n
X
k=1
zk
Qn
¡
¢
2 exp [−z |t|]
k
l=1,l6=k zl
¡ 2
¢
Qn
,
2
l=1,l6=k −zk + zl
On the distributions of the densities involving non-zero ...
169
In the similar way we have
Z ∞
itx
e
−∞
¡ 2¢
Qn
Qn ¡ 2 ¢
n
X
exp [zk t]
zk
l=1,l6=k zl
k=1
¢ dx = πλ1
¡ 2
¢ ,
Qn ¡ 2
zk Qn
2
x +z
−z + z 2
λ1
k=1
when t < 0.
(2.5)
Z ∞
itx
e
−∞
k
k
l
¡ 2¢
Qn
Qn ¡ 2 ¢
n
X
zk
l=1,l6=k zl
k=1
¢ dx = πλ1
¡ 2
¢,
Qn ¡ 2
zk Qn
x + z2
−z + z 2
λ1
k=1
when t = 0.
(2.6)
l=1,l6=k
k=1
k
k=1
l=1,l6=k
k
l
Therefore, on combining the equations (2.4a), (2.4b) and (2.4c) we evaluate
Z ∞
itx
e
−∞
¡ 2¢
Qn
Qn ¡ 2 ¢
n
X
exp [zk |t|]
zk
l=1,l6=k zl
k=1
¢
¡ 2
¢ ,
Qn ¡ 2
zk Qn
2 dx = πλ1
2
λ1
k=1
x + zk
l=1,l6=k
k=1
−zk + zl
−∞ < t < ∞.
(2.7)
Hence, the theorem has been proved.
2.1. Related Functions and Polynomials:
Set u = e−|t| (i.e. |t| = logu−1 ), 0 < u < 1 in (2.2), we find a function
(2.8)
Φ1 (u) = πλ1 u
n
X
zk
k=1
Again, set πλ1 u =
Q1 (u) =
(2.9)
Qn−1
2
2
Qnl=1 (−zn +zl ) in (2.6), we find another
z
(−1)n−1 zn
l=1 l
function
(−1)n−1 zn
¡ 2¢
z −1
l=1,l6=k zl (u) k
¡ 2
¢
Qn
2 .
l=1,l6=k −zk + zl
Qn
¡ 2¢
Qn
¢ n
Qn−1 ¡ 2
zk−1
−zn + zl2 X
l=1,l6=k zl (u)
l=1
¡ 2
¢
Qn
zk Qn
2
l=1 zl
k=1
l=1,l6=k
−zk + zl
170
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
Now, set zl = l, l = 1, 2, ..., n, in (2.7), we find a polynomial such that
(n−1)
(2.10)
Q1
n
X
(n − 1)! (2n)!
k (−u)k−1 .
(n
−
k)!
(n
+
k)!
k=1
(u) =
For the probability density (1.5) the Takanos polynomial [10] is given
by
Pn−1 (u) =
(2.11)
n
X
(2n)!
k
(−u)k−1 , n ≥ 1.
(n − k)! (n + k)! n
k=1
Then, make an appeal to the eqns. (2.8) and (2.9) to get a relation
(n−1)
Q1
(2.12)
(u) = n!P(n−1) (u), n ≥ 1.
Thus due to the relation (2.10) and the analysis given in [10, section-2,
eqns. (2.3)-(2.8)] we may guess the curve of the trigonometric sum which
has non- zero zeros (in form of natural
numbers) of Bessel function.
Qn−1
2
2
(−1)n−1 zn
(
l=1
Qn 2 −zn +zl ) in (2.6) and then put
Further, set πλ1 u =
z
l=1 l
zl = 1l , ∀l = 1, 2, ..., n, to find a function
(n−1)
R1
(u) =
n
X
(k)2n−1
2n−1
k=1
(n)
(2n)!
(−uk )k−1 , where,
(n − k)! (n + k)!
uk = u(−1/k) , 0 < u < 1, n ≥ 1.
(2.13)
The proof of (2.11)
follows readily from the formula (2.6) by setting
Qn−1
2
2
(−1)n−1 zn
Qn l=12 (−zn +zl )
πλ1 u =
z
l=1 l
and then putting zl = 1l , ∀l = 1, 2, ..., n, to get
(n−1)
R1
n−1
(u) = (−1)
n−1
Y
l=1
Ã
n2 − l2
n2
! n µ ¶
X n
k=1
k
n
Y
l=1,l6=k
Ã
k2
2
k − l2
(2.14)
Then from the equation (2.12), we easily obtain (2.11).
If we suppose that
!
(u)1/k
−1
On the distributions of the densities involving non-zero ...
(n−1)
S1
((n−1)
(u) = uR1
((n−1)
(u), S1
171
(u) is many times differentiable,
n ≥ 1,
(2.15)
and
Hn−1 (u) =
n
X
k=1
(2n)! (−1)k−1
(n − k)! (n + k)!
µ ¶
1
k
(u) k , n ≥ 1
n
Then with the aid of (2.11), (2.13) and (2.14), we find that
(2.18)
µ
d
u
du
¶2n−2
(n−1)
S1
(u) =
1
2n−2
(n)
Hn−1 (u) , n ≥ 1.
Thus due to the relation (2.15) and the analysis given in [10, section-2,
eqns. (2.3)-(2.8)] we may guess the curve of the trigonometric sum, this
curve has non-zero zeros (in form of rational numbers) of Bessel function.
3. The Distributions of the Characteristic Function Involving Non-zero Zeros of Legendre Function and its Related
Polynomials
Theorem-2 : If in the complex upper half plane the characteristic function
for the density function (1.9) exists by the formula
(3.1)
Z ∞
Φ2 (t) =
eitx
−∞
¡
¢
Qn
1 − ςk2
¡
¢
Qn k=1
dx, −∞ < t < ∞,
2
2
λ2
k=1
x + ςk
then, there holds the distribution formula of non-zero real zeros lying between −1 to +1, ςk , (i.e.ςk 6= 0, and −1 < ςk < 1), ∀k = 1, 2..., n, of Legendre function on the real curve of t such that
Φ2 (t) = πλ2
n ³
X
ςk−1
k=1
(3.2)
− ςk
´ Qn
¡
¢
2
l=1,l6=k 1 − ςk exp [−ςk
¡ 2
¢
Qn
2
l=1,l6=k −ςk + ςl
|t|]
, −∞ < t < ∞,
Proof : In the similar manner, as analyzed to prove theorem-1, we
prove theorem-2.
172
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
3.1. Related Functions and Polynomials:
Set u = e−|t| (i.e. |t| = logu−1 , 0 < u < 1 in (3.2), to find a function
n ³
X
ςk−1
Φ2 (u) = πλ2 u
k=1
¡
¢
2 (u)ςk−1
´ Qn
l=1,l6=k 1 − ςl
¡ 2
¢ , ςk 6= 0
Qn
− ςk
2
l=1,l6=k −ςk + ςl
and −1 < ςk < 1, ∀ k = 1, 2...
(3.3)
Again, set πλ2 u =
Qn−1 2 2
Qn l=1 (ςl −ςn ) in (3.3), to find a function
(1−ς )
(−1)n−1
l
l=1
n−1
Q2 (u) = (−1)
n−1
Y
l=1
³
ςl2
− ςn2
and −1 < ςk < 1
(3.4)
Qn
n
´ X
1
(1 + ςl ) (u)ςk−1
¡ 2
¢
Ql=1
n
2 , ς 6= 0,
k=1
ςk
−ςk + ςl
l=1,l6=k
Now, set ςl = l, l = 1, 2, ..., n, in (3.4), we find a polynomial such that
(n−1)
Q2
(u) =
(3.5)
n
(2n)!
k
(n + 1)! X
(−u)k−1 , n ≥ 1, ∀k = 1, 2, ...
n
n
(n
−
k)!
(n
+
k)!
k=1
Now, with the aid of (2.9), and (3.5) we find that
(n + 1)!
Pn−1 , n ≥ 1.
n
Thus due to the relation (3.6) and the analysis given in [10, section-2,
eqns. (2.3)-(2.8)], we may guess the curve of the trigonometric sum which
has no non- zero zeros (in form of natural numbers) of Legendre function.
Again, put ςl = 1l , ∀l = 1, 2, ..., n, in (3.4), we find that
(n−1)
Q2
(3.6)
(n−1)
R2
(3.7)
(u) =
n
Y
n−1
(u) = (−1)
Ã
n2 − l2
−1
n2 l2
l=1
´
−1
Qn ³
! n
(u)1/k
X k l=1 l+1
l
³ 2 2´
Qn
k −l
k=1
l=1,l6=k
k 2 l2
On the distributions of the densities involving non-zero ...
173
Then on solving (3.7) we easily obtain
(n−1)
R2
(u) =
µ
¶ n µ ¶2n−1
n+1 X
k
(2n)!
(−u)k−1 ,
n
n
(n
−
k)!
(n
+
k)!
k=1
where, uk = u−1/k , 0 < u < 1.
(3.8)
Now we suppose that
(n−1)
S2
(n−1)
(u) = uR2
(n−1)
(u) , S2
(u)
is many times differentiable, n ≥ 1. (3.9)
Then we find the relation
(3.10)
µ
d
u
du
¶2n−2
(n−1)
S2
(u) =
(n + 1)
Hn−1 (u) , n ≥ 1.
(n)2n−1
The function Hn−1 (u) is defined in the equation (2.14).
From (3.10) we may guess the curve on which the distributions of nonzero real zeros lie between −1 to +1 of Legendre function.
4. On the Infinite Divisibility
Theorem-3 : The distributions, with the probability densities (1.8) and
(1.9), are infinite divisible and thus (1.6) and (1.7) are also infinite divisible.
Proof : (A) : From (1.8), we write the equality (See Takano [10], p.3)
Qn
P
(z2 )
Q
f1 (x) = λ1 n k=1 2 k 2 = λ1 nk=1 Qn
Qn
z2
l=1 l
2
−zk +zl2
l=1,l6=k
( )
,
x
+z
(
)
(
)(x2 +zk2 )
k
k=1
−∞ < x < ∞, zk 6= 0, ∀ k = 1, 2, ..., n.
(4.1)
174
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
Again, using the relation
1
=
2
x + zk2
Z ∞
0
2
1
−x2 /v √ −zk /v
√
πe
(v)α−2 dv, α > 1
α e
π (v)
and α 6= 2, 3, ..., zk 6= 0, in (4.1) we find that
f1 (x) =
Z ∞
0
¡ ¢
√ Q
n
X
1
λ1 π nl==1 zl2 −z2 /v
−x2 /v
¡
¢ e k (v)α−2 dv, α > 1
√
Qn
α e
2
2
π (v)
l=1,l6=k −zk + zl
k=1
α > 1 and α 6= 2, 3, ..., zk 6= 0, ∀k = 1, 2, ..., n.
(4.2)
Now, from (4.2) we set that
n
√ X
Qn
g1 (v) = λ1 π
k=1
(4.3)
Qn
l==1
¡
l=1,l6=k
¡ 2¢
z
l
−zk2 + zl2
2
¢ e−zk /v (v)α−2
α > 1 and α 6= 2, 3, ..., zk 6= 0, ∀k = 1, 2, ..., n, v > 0.
When n = 1, in (4.3), we have
√
2
g1 (v) = λ1 πz12 e−z1 /v (v)α−2 , z1 6= 0, α > 1
and α 6= 2, 3, ..., v > 0
(4.4)
When n = 2, we have
"
#
√
z2
z2
2
2
g1 (v) = λ1 π ¡ 2 1 2 ¢ e−z1 /v + ¡ 2 2 2 ¢ e−z2 /v (v)α−2 , z1 , z2 6= 0, α > 1
−z1 + z2
−z1 + z2
and α 6= 2, 3, ..., v > 0
(4.5)
On the distributions of the densities involving non-zero ...
175
In the above results (4.3)-(4.5) the distribution g1 (v) is of type normal
distribution and has the finite variables (z1 , ..., zn ) and is continuous to
the variable v so that it is continuous and partially discrete thus it is a
mixing density function of normal distribution and it is positive for α > 1
and α 6= 2, 3, ..., v > 0. Again the distribution with the density g1 (v) is
infinitely divisible and hence for that following relations hold:
− χ01 (s)
(4.6)
where
(4.7)
= χ1 (s)
χ1 (s)
Z ∞
0
Z ∞
0
e−st k1 (t) dt,
e−sv g1 (v) dv
The k1 (t) is a non-negative bounded function given by
(4.8)
k1 (t) = lim
R→∞
1
2πi
Z ξ+iR
ets
ξ−iR
µ
−
¶
χ01 (s)
ds, ξ > 0, t > 0.
χ1 (s)
Then, make an appeal to the equations (4.4) and (4.7) we easily find
that
∞
X
√
Γ (α − m − 1) ³ 2 ´m
χ1 (s) = λ1 πz12
−z1
(s)m−α+1 , α > 1
m!
m=0
and α 6= 2, 3, ...
(4.9)
Then from (4.9) we have
∞
X
√
Γ (α − m) ³ 2 ´m
−z1
(s)m−α , α > 1
χ01 (s) = −λ1 πz12
m!
m=0
and α 6= 2, 3, ...(4.10)
Thus from the equations (4.8), (4.9) and (4.10), we obtain
1
k1 (t) = lim
R→∞ 2πi
Z ξ+iR
ξ−iR
ets
à P∞
s
α > 1 and α 6= 2, 3, ..., ξ > 0, t > 0
(4.11)
!
¢
Γ(α−m) ¡
2 m (s)m−α
−z
m=0
1
m!
ds,
P∞ Γ(α−m−1) ¡ 2 ¢m
−z1
(s)m−α
m=0
m!
176
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
Check the following inequalities for all m = 0, 1, 2, ..., and α 6= 2, 3, ...
α + m + 1 > α + m ⇒ Γ (α − m) > Γ (α − m − 1), and
(4.12)
α < α + m + 1 ⇒ Γ (α − m) < αΓ (α − m − 1)
Then, from the inequalities given in (4.12) we easily obtain
1
<
s
(4.13)
Ã
s
!
¢m
Γ(α−m) ¡
−z12
(s)m−α
m=0
m!
P∞ Γ(α−m−1) ¡ 2 ¢m
−z1
(s)m−α
m=0
m!
P∞
<
α
s
Then, with the help of (4.11) and (4.13) we easily obtain
(4.14) 0 < k1 (t) < (α − 1) , α > 1, and α 6= 2, 3, ..., ∀t ∈ (−∞, ∞)
(Since k1 (t) is independent of t.)
Thus the relations (4.6)-(4.8) hold and k1 (t) is non-negative for
α > 1,and α 6= 2, 3, ..., ∀t ∈ (−∞, ∞) and thus the density function g1 (v)
is infinitely divisible (See also [3], [8], [9], [10] [11]) and therefore the distribution with density function (1.8) is also infinitely divisible.
Again, there exists
(4.15)
Qn
lim f1 (x) = lim λ1 Qn
n→∞
n→∞
k=1
¡
k=1
¡ 2¢
z
k
x2
+ zk2
¢ →
λ
Q∞
k=1
µ1
x2
zk2
1+
¶
where γ1 is an arbitrary constant, −∞ < x < ∞, and zk 6= 0, ∀k = 1, 2....
Therefore, the distribution with density (1.6) is also infinitely divisible.
The probability distribution with density (1.6) converges in the sense:
(See, [7], [10])
Φ1 (t) → Ψ1 (t) =
Z ∞
eitx
−∞
λ
Q∞
k=1
µ1
1+
x2
zk2
¶ dx, λ1
is an arbitrary constant, −∞ < t < ∞, and zk 6= 0, ∀ k = 1, 2...
(4.16)
(B)- In the similar manner, from (1.8) we write the equality
¢
Qn ¡
n
2
X
k=1¡ 1 − ςk ¢
Qn
=
λ
f2 (x) = λ2 Qn
2
2
2
k=1
x + ςk
k=1
¡
¢
Qn
1 − ςk2
t==1
¡ 2
¢¡
2
l=1,l6=k
−ςk + ςl
x2 + ςk2
¢,
On the distributions of the densities involving non-zero ...
177
−∞ < x < ∞, ςk 6= 0 and −1 < ςk < 1, ∀ k = 1, 2, ...
(4.17)
Again, with the help of the relation given in (4.2) and (4.17) we find
f2 (x) =
R∞
0
2
e−x /v
Pn
√ Qn
λ2 π l==1 (1−ςl2 )
2
e−ςk /v (v)α−2 dv, α > 1
−ςk2 +ςl2 )
l=1,l6=k (
6 0 and − 1 < ςk < 1, ∀ k = 1, 2, ...
and α 6= 2, 3, ..., ςk =
1
π(v)α
√
k=1
Qn
(4.18)
Now, from (4.18) we set that
¡
¢
Qn
n
√ X
1 − ς2
2
Qn l==1 ¡ 2 l 2 ¢ e−ςk /v (v)α−2 ,
g2 (v) = λ2 π
l=1,l6=k −ςk + ςl
k=1
α > 1 and α 6= 2, 3, ..., ςk 6= 0 and −1 < ςk < 1, ∀ k = 1, 2, ..., v > 0
(4.19)
When n = 1 in (4.19), we have
´
√ ³
2
g2 (v) = λ2 π 1 − ς12 e−ς1 /v (v)α−2 , ς1 6= 0, −1 < ς1 < 1, α > 1
and α 6= 2, 3, ..., v > 0
(4.20)
When n = 2 in (4.19), we have
"
#
√
1 − ς2
1 − ς2
2
2
g2 (v) = λ2 π ¡ 2 1 2 ¢ e−ς1 /v + ¡ 2 1 2 ¢ e−ς2 /v (v)α−2 , ς1 , ς2 6= 0
−ς1 + ς2
−ς2 + ς1
(4.21)
and −1 < ς1 < 1, −1 < ς2 < 1, α > 1, and α 6= 2, 3, ..., v > 0.
178
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
The above results (4.19)-(4.21) show that g2 (v) is a mixing probability
density function of normal distribution and it is positive for α > 1 and
α 6= 2, 3, ..., v > 0. Again the distribution with the density g2 (v) is infinitely
divisible and for that following relations hold:
Z ∞
− χ02 (s) = χ2 (s)
(4.22)
0
e−st k2 (t) dt,
where
(4.23)
Z ∞
χ2 (s) =
e−sv g2 (v) dv.
0
The k2 (t) is a non-negative bounded function and there exists
1
k2 (t) = lim
R→∞ 2πi
Z ξ+iR
ξ−iR
ts
e
µ 0
¶
χ2 (s)
χ2 (s)
ds, ξ > 0, ∀t ∈ (−∞, ∞).
(4.24)
From the techniques applied in (4.9)-(4.14), we examine and conclude
that k2 (t) is non-negative and bounded function for α > 1 and α 6=
2, 3, ..., ∀t ∈ (−∞, ∞) and thus the density function g2 (v) is infinitely divisible (See also [3], [8], [9], [10] [11]) and therefore the distribution with
density function (1.9) is also infinitely divisible.
Again there exists
(4.25)
¡
¢
Qn
2
k=1 ¡ 1 − ςk ¢
lim f2 (x) = lim λ2 Qn
2 →
2
n→∞
n→∞
k=1
x + ςk
λ2
Q∞
k=1
µ
x2 +ςk2
1−ςk2
¶,
where γ2 is an arbitrary constant, −∞ < x < ∞, and ςk 6= 0 and −1 <
ςk < 1, ∀k = 1, 2... Therefore, the distribution with density (1.7) is also
infinitely divisible.
The probability distribution with density (1.7) converges in the sense:
(See, [7], [10])
Φ2 (t) → Ψ2 (t) =
Z ∞
−∞
eitx
λ
Q∞
k=1
µ2
x2 +ςk2
1−ςk2
¶ dx, λ2
On the distributions of the densities involving non-zero ...
179
is an arbitrary constant, −∞ < t < ∞, and ςk 6= 0, −1 < ςk < 1,
∀ k = 1, 2, ...
(4.26)
is an arbitrary constant, −∞ < t < ∞, and ςk 6= 0, −1 < ςk < 1, ∀k = 1, 2...
5. Discussions
Takano [10, section-2, eqn. (2.4)] has taken a polynomial
(5.1) h (u) =
n
2n!
d
1 X
uk , h (u) = −Pn−1 (u)
(−1)k
n k=1
(n − k)! (n + k)!
du
Then, from (5.1) for u = eiθ , (0 ≤ θ ≤ 2π) and using following identities
n
X
k
(−1)
k=1
µ ¶
2n!
θ
1
cos kθ = 22n−1 sin2n
−
(n − k)! (n + k)!
2
2
Ã
2n
n
!
,
(5.2)
and
n
X
(−1)k
k=1
³
22r sin θ2
(5.3)
´2r
n−1
X (2n − 2r − 2)!
2n!
sin kθ = − sin θ
2
(n − k)! (n + k)!
r=0 {(n − r − 1)!}
Takano [10] has guessed the curve of trigonometric sum given in the
equations (5.2) and(5.3).
Now consider a polynomial
(5.4)
(n)
D1
(u) =
n
X
k=1
(−1)k
(2n)! (n − 1)!
uk
(n − k)! (n + k)!
180
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
Then by equations (2.9), (2.10) and (5.4) we have
d
(n)
(n−1)
D
(u) = − (n)! Pn−1 (u) = −Q1
(u) , n ≥ 1.
du 1
(5.5)
Now in (5.4) take u = eiθ , (0 ≤ θ ≤ 2π) and use the identities (5.2) and
(5.3) and³suppose
that
´
(n)
iθ
D1
e
= U (θ) + iV (θ) we get
h
³ ´
i
U (θ) = n1 22n−1 (n)! sin2n θ2 − 12 (2n)!
(n)!
and
³
´2r
P
(2n−2r−2)! 2r
θ
2
sin
, n≥1
V(θ) = − (n − 1)! sin θ n−1
r=0 {(n−r−1)!}2
2
Thus from right hand side of (5.6) we may guess the curve whose slope
has non-zero zeros of Bessel function.
Further put u = eimθ , (0 ≤ θ ≤ 2π
m , m > k) in (2.14) we get
³
imθ
Hn−1 e
+i
Pn
´
"
µ
¶
n
(2n)! (−1)k k
m
1 X
cos
θ
=−
n k=1 (n − k)! (n + k)!
k
(2n)! (−1)k k
k=1 (n−k)! (n+k)!
sin
¸
¡m ¢
k
θ
, n ≥ 1, (5.6)
At θ = 0 from (5.7) we have
(5.7)
³
´
Hn−1 eimθ =
(2n − 2)!
, n≥1
(n − 1)! (n)!
Thus using the relations (2.15) and (3.10) in (5.7) and (5.8) respectively
we may find the curves which have the non-zero zeros (in rational form) of
Bessel and Legendre functions. Due to lack of space we omit them.
References
[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publication, New York, (1970).
[2] L. Bondesson, On the infinite divisibility of the half-Cauchy and other
decreasing densities and probability functions on the non-negative line,
Scand. Acturial J., pp. 225-247, (1985).
On the distributions of the densities involving non-zero ...
181
[3] M. J. Goovaerts, L. D Hooge and N. De, Pril, On the infinite divisibility
of the product of two Γ-distributed stochastically variables, Applied
Mathematics and Computation, 3, pp. 127-135, (1977).
[4] H. Hochstadt, The Functions of Mathematical Physics, New York,
Dover, (1986).
[5] D. H. Kelkar, Infinite divisibility and variance mixtures of the normal
distribution, Ann. Math. Statistic. 42, pp. 802-808, (1971).
[6] H. Kumar and S. Srivastava, On some sequence of integrals and their
applications, Bull. Cal. Math. Soc. 100 (5), pp. 563-572, (2008).
[7] K. Sato, Class L of multivariate distributions and its sub classes, J.
Multivaria. Anal. 10 (1980), pp. 207-232, (1980).
[8] F. W. Steutel, Preservation of infinite divisibility under mixing and
related topics, Math. Center, Tract. Amsterdam, 33, (1970).
[9] K. Takano, On a family of polynomials with zeros outside the unit disk,
Int. J. Comput. Num. Anal. Appl. 1 (4), pp. 369-382, (2002).
[10] K. Takano, On the infinite divisibility of normed conjugate product of
Gamma function, Proc. of 4th Int. Conf. SSFA, (4), pp. 1-8, (2003).
[11] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand.
Acturial. J. pp. 31-40, (1977).
Hemant Kumar
Department of Mathematics,
D. A-V. P. G., College,
Kanpur, U. P.
India
e-mail : palhemant2007@rediffmail.com
M. A. Pathan
Department of Mathematics,
University of Botswana,
Private Bag 0022,
Gaborone,
Botswana
e-mail : [email protected]
182
Hemant Kumar, M. A. Pathan and R. C. Singh Chandel
and
R. C. Singh Chandel
Department of Mathematics,
D. V. P. G., College,
Orai, U. P.,
India
e-mail : [email protected]