Proceedings of the V Congress of Mathematicians of Macedonia,
September 24–27, 2014, Ohrid, R. Macedonia
Vol.2 (2015) 27–32
c
2015
Union of Mathematicians of Macedonia
PROBABILITY DISTRIBUTIONS ASSOCIATED WITH
GENERALIZATION OF THE PLANCK’S LAW
DELČO LEŠKOVSKI
Abstract. The main object of this paper is to present a study of probability
density functions and distributions associated with a generalized Planck’s law.
Characteristic functions and fractional moments related to the probability
density functions of the considered distributions are derived by means of
generalized Hurwitz-Lerch Zeta function introduced in [4].
1. Introduction and Preliminaries
The following familiar infinite series
∞
X
2n
S(r) =
2 + r 2 )2
(n
n=1
(r ∈ R+ )
is named after Emile Leonard Mathieu (1835 − 1890), who investigated it in his
1890 work [5] on elasticity of solid bodies.
The generalized Mathieu series
∞
X
2aβn
r, α, β, µ ∈ R+
(1)
Sµ(α,β) (r, a) = Sµ(α,β) (r, {an }∞
)
=
n=1
α
2 )µ
(a
+
r
n
n=1
is introduced by Srivastava and Tomovski in [7], where it is tacitly assumed that
the positive sequence a := {a(n)}n∈N0 ( lim an = ∞) is so chosen (and then the
n→∞
positive parameters α, β and µ are so constrained) that the infinite series (1)
∞
P
1
converges, that is, that the auxiliary series
is convergent.
aµα−β
n=1
n
The Mathieu (a, λ)-series
S(q, p, a, λ) =
∞
X
a(n)
p
(λ(n)
+ q)
n=0
q, p ∈ (0, ∞)
is introduced by Pogany in [6], assuming that the sequences a := {a(n)}n∈N0 ,
λ := {λ(n)}n∈N0 are positive, real and λ monotonously increases to ∞.
2010 Mathematics Subject Classification. 33B15, 33E20, 60E10.
Key words and phrases. Generalized Mathieu series, generalized Hurwitz-Lerch Zeta function,
characteristic function, fractional moment of order s, generalized Planck’s distribution.
27
28
D. LEŠKOVSKI
The generalized Hurwitz-Lerch Zeta function Φ (z, s, a) is defined e.g in (Erdelyi
at al., [3, p.27, Eq.1.11.1]) as the power series
Φ (z, s, a) =
∞
X
zn
s,
(n + a)
n=0
where a ∈ C \ Z−
0 ; < (s) > 1 when |z| = 1 and s ∈ C when |z| < 1 and continues meromorphically to the complex s-plane, except for the simple pole at s = 1,
with its residue equal to 1. One special case of the Hurwitz-Lerch Zeta function Φ (z, s, a) is generalized Hurwitz-Lerch Zeta function Φ∗α (z, s, a) (Goyal and
Laddha, [4]), defined as a power series
Φ∗α (z, s, a) =
∞
X
(α)n
zn
,
n! (n + a)s
n=0
where a ∈ C \ Z−
0 , s ∈ C when |z| < 1 and < (s − α) > 0 when |z| = 1.
Remark 1.1: It is obvious that Φ∗1 (z, s, a) = Φ (z, s, a) and Φ∗1 (1, s, 1) = ζ (s),
where ζ stands for the Riemann Zeta function.
The Planck’s law describes the electromagnetic radiation emitted from a black
body at absolute temperature T . As a function of frequency v, Planck’s law represents the emitted power per unit area of emitting surface in the normal direction,
per unit solid angle, per unit frequency. Planck’s law is sometimes written in terms
of the spectral energy density per unit volume of thermodynamic equilibrium cavity radiation (Brehm and Mullin, [1]). Then, the frequency spectral density is
given by
8πhv 3
1
,
v>0
c3 ehv/kT − 1
where constants h, c, k possess the physical meanings as Planck constant, speed
of light and Boltzmann constant respectively.
Having in mind the well known identity
Z ∞ z−1
x
dx = α−1 ζ (z) Γ (z) α, < (z) > 0
αx
e −1
0
u (v) =
where ζ, Γ stand for the Riemann Zeta and Gamma function respectively, Tomovski
et al. recognized (Tomovski et al., [8]) that the random variable X defined on some
fixed standard probability space (Ω, F, P), possessing PDF (probability density
function)
αp+2
xp+1
· αx
· χR+ (x)
ζ (p + 2) Γ (p + 2) e − 1
is distributed according to Planck’s law with parameter p > −1. They wrote that
correspondence as X ∼ P lanck (p). Symbol χR+ (x) denotes the indicator function
of the set S.
u (x) =
PROBABILITY DISTRIBUTIONS ASSOCIATED WITH ...
29
The related CDF (cumulative distribution function) reads
(
R x tp+1
αp+2
dt x > 0
U (x) = ζ(p+2)Γ(p+2) 0 eαt −1
0
x≤0
They were interested in the characteristic function (CHF φ (r) = EeirX , r ∈ R)
and the fractional moment of order s (EX s ), for the random variable X having
Planck distribution. Among the other results, Tomovski et al. in [8] gave the
following theorems and example:
Theorem 1.1. ([8], Theorem 2.1) The CHF associated with the random variable
X ∼ P lanck (p) reads, as follows
1
ir
φ (r) =
Φ 1, p + 2, 1 −
, r ∈ R.
ζ (p + 2)
α
Moreover, there holds the estimate
r
1
(2,0)
|φ (r)| ≤
Sp/2+1
; {k} ,
2ζ (p + 2)
α
r ∈ R.
Theorem 1.2. ([8], Theorem 2.3) Let X ∼ P lanck (p) , p > −1. Then the fractional moment EX s of order s is given by
EX s =
Γ (s + p + 2) ζ (s + p + 2)
,
as Γ (p + 2) ζ (p + 2)
s > −p − 1.
Example 1.1: ([8], Example 2.6) Consider the random variable Xg having PDF
x
12
χR (x) ;
g (x) = 2 · x
π e +1 +
The corresponding CDF will be
Z x
12
t
dt.
G (x) = 2 χR+ (x)
t
π
0 e +1
The characteristic function expressed via Hurwitz-Lerch Zeta becomes
12
φg (r) = 2 Φ (−1, 2, 1 − ir) .
π
The main objective of this talk is to introduce a probability distributions which
are generalizations of the Planck distribution and the probability distribution given
in Example 1.4 and to express their corresponding PDF and CDF in terms of the
Gamma function and the generalized Hurwitz-Lerch Zeta function Φ∗α (z, s, a).
2. Probability distributions associated with
one generalization of the Planck’s law
This work is based on the work of Carvalho and Souza. In [2] they obtained
new physical systems where the density of the energy per unit volume is given by
Ωd (v) =
8πHv 3
d ,
z d−1 c3 eβHv/d − 1
v > 0,
(2)
30
D. LEŠKOVSKI
where v denotes frequency; c-speed of light; H = 2ς, ς-the initial action per
1
1
, 1 < α ≤ 2, 1 ≤ d < ∞; β = kT
, k-Boltzmann constant,
oscillator; d = α−1
T -temperature; z-pure dimensionless number.
Remark 2.1: The formula (2) is a generalization of the Planck’s law. For d = 1,
H ≡ h, where h is the Planck constant.
Lemma 2.1. Let α, p > 0; 0 < β < p.Then the following integral formula holds
true
Z ∞
xp−1
Γ (p) ∗
Φ (1, p, β)
(3)
dx =
β
αx
αp β
(e − 1)
0
Proof. Applying binomial expansion and integral representation for gamma function, we obtain:
R∞
xp−1
dx
(eαx −1)β
R∞
−β
xp−1 e−αβx (1 − e−αx ) dx
R ∞ p−1 −αβx P∞
−β
−αx n
= 0 x e
(−e
) dx
n=0
n
h
i
R∞
P∞ (β)n −αnx
= 0 xp−1 e−αβx
dx
n=0 n! e
0
=
P∞
=
P∞
=
Γ(p) ∗
α p Φβ
n=0
n=0
(β)n
n!
=
R∞
0
0
xp−1 e−α(n+β)x dx
(β)n
Γ(p)
n! [α(n+β)]p
(1, p, β).
=
Γ(p)
αp
P∞
n=0
(β)n
1
n! (n+β)p
Bearing in mind (3), we recognize that the random variable X, defined on some
fixed standard probability space (Ω, F, P), possessing PDF
αp
xp−1
· χR+ (x) , α > 0, p > 1; 1 ≤ β < p
·
Φ∗β (1, p, β) Γ (p) (eαx − 1)β
is distributed according the generalized Planck’s law with parameter p > 1, given
by (2). We write this correspondence as X ∼ GP (p; β).
The related CDF reads
(
R x tp−1
αp
x>0
∗
β dt
Ωβ (x) = Φβ (1,p,β)Γ(p) 0 (eαt −1)
0
x≤0
ωβ (x) =
Theorem 2.1. The CHF associated with the random variable X ∼ GP (p; β) reads
as follows
Φ∗β 1, p, β − ir
α
φ (r) =
,
r ∈ R.
(4)
Φ∗β (1, p, β)
Moreover, there holds the estimate
1
r 2 p (β)k
|φ (r)| ≤ ∗
S
, ,
, (k + β)2
,
r∈R
(5)
Φβ (1, p, β)
α
2
k!
PROBABILITY DISTRIBUTIONS ASSOCIATED WITH ...
31
Proof. By direct calculations we have
φ (r) = EeirX =
=
αp
Φ∗
β (1,p,β)Γ(p)
=
1
Φ∗
β (1,p,β)
αp
Φ∗
β (1,p,β)Γ(p)
P∞
P∞
0
0
R∞
(β)n
n!
0
R∞
0
p−1
x
eirx (eαx
dx
−1)β
xp−1 e−((n+β)α−ir)x dx
(β)n
1
p
n! (n+β− ir
α )
which proves (4) being the generalized Hurwitz-Lerch Zeta series convergent
(p > β).
The estimate (5) follows from
P∞ (β)n
1
1
|φ (r)| = Φ∗ (1,p,β)
ir
n=0 n! (n+β− )p β
α
≤
1
Φ∗
β (1,p,β)
P∞
=
1
Φ∗
β (1,p,β)
P∞
(β)n
1
n=0 n! |n+β− ir |p
α
n=0
(β)n
1
n! ((n+β)2 +( r )2 )p/2 .
α
We point out that φ(0) = 1.
Theorem 2.2. Let X ∼ GP (p; β). Then the fractional moment EX s is given by
Γ (p + s) Φ∗β (1, p + s, β)
,
αs Γ (p) Φ∗β (1, p, β)
EX s =
Proof. EX s =
αp
Φ∗
β (1,p,β)Γ(p)
=
Γ(p+s) ∗
αp
p+s Φβ
Φ∗
β (1,p,β)Γ(p) α
=
Γ(p+s)Φ∗
β (1,p+s,β)
αs Γ(p)Φ∗
β (1,p,β)
R∞
0
s > −p.
(6)
xp+s−1
dx
(eαx −1)β
(1, p + s, β)
.
Remark 2.2: Theorem 1.1 and Theorem 1.2 are corollaries of Theorem 2.1 and
Theorem 2.2, respectively (β = 1 and Remark 1.1).
Example 2.1: Now, we consider the random variable X ∼ GP (4; d) , 1 < d < 4,
H
where α = kT
d , which corresponds to the generalized Planck’s distribution. The
related PDF is
ωd (x) =
α4
x3
·
· χR+ (x) ,
6Φ∗d (1, 4, d) (eαx − 1)d
formula (4) reduces to
Φ∗d 1, 4, d − ir
α
φ (r) =
,
Φ∗d (1, 4, d)
r ∈ R,
32
D. LEŠKOVSKI
and formula (6) reduces to
EX s =
Γ (s + 4) Φ∗d (1, s + 4, d)
,
6αs Φ∗d (1, 4, d)
s > −4.
Example 2.2: Bearing in mind the result
Z ∞
xp−1
Γ (p) ∗
Φ (−1, p, β) ,
α, p > 0; 0 < β < p,
dx =
β
αx
αp β
(e + 1)
0
we can define a random variable Xg possessing PDF
αp
xp−1
· χR+ (x) ,
·
(−1, p, β) Γ (p) (eαx + 1)β
The corresponding CDF is
(
R x p−1
p
g (x) =
α > 0, p > 1, 1 ≤ β < p
Φ∗β
G (x) =
t
dt
0 (eαt +1)β
α
Φ∗
β (1,p,β)Γ(p)
0
x>0
x≤0
The characteristic function is
Φ∗β −1, p, β − ir
α
φg (r) =
,
Φ∗β (−1, p, β)
while
EXgs =
Γ (p + s) Φ∗β (−1, p + s, β)
,
αs Γ (p) Φ∗β (−1, p, β)
r ∈ R,
s > −p.
References
[1] J. J. Brehm, W. J. Mullin, Introduction to the Structure of Matter: A Course in Modern
Physics, Wiley, New York, 1989.
[2] R. D. B. Carvalho, A. M. C. Souza, Generalizin Planck’s distribution by using the CaratiGalgani model of molecular collisions, Physica A Statistical Mechanics and its Applications
Vol. 391, issue 9, May (2012), 2911–2916.
[3] A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi (eds.), Higher Transcendental
Functions I, II, McGraw-Hill Book Comp., New York, Toromto and London, 1953.
[4] S. P. Goyal, R. K. Laddha, On the generalized Zeta function and the generalized Lambert
function, Ganita Sandesh, 11 (1997), 99–108.
[5] E. L. Mathieu, Traite de physique mathematique, Vol. VI-VII, part 2, Paris, 1890.
[6] T. K. Pogany, Integral representation of Mathieu (a, λ)-series, Integral Transforms Spec.
Funct. 16 (2005), no.8, 685–689.
[7] H. M. Srivastava, Ž. Tomovski, Some problems and solutions involving Mathieu series and
its generalizations, JIPAM, 5(2) (2004), Art. 45.
[8] Ž. Tomovski, R. K. Saxena, T. K. Pogany, Probability distributions associated with Mathieu
type series, Probab. Stat. Forum Vol 5, July (2012), 86–96.
International Balkan University, Faculty of Engineering,
Tashko Karadza 11A, 1000 Skopje, Republic of Macedonia
E-mail address: [email protected], [email protected]