#### Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications

International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporation
Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications
M. Aslam Chaudhry
Asghar Qadir
Recommended by Virginia Kiryakova
Fermi-Dirac and Bose-Einstein functions arise as quantum statistical distributions. The Riemann zeta function and its extension, the polylogarithm function, arise in the theory of numbers. Though it might not have been expected, these two sets of functions belong to a wider class of functions whose members have operator representations. In particular, we show that the Fermi-Dirac and Bose-Einstein integral functions are expressible as operator representations in terms of themselves. Simpler derivations of previously known results of these functions are obtained by their operator representations.
1. Introduction
The study of analytic functions is very useful for the application of mathematics to
various physical and engineering problems and for the development of a further
understanding of mathematics itself. In particular, the Riemann zeta function [1, page 1]
has played an important role in number theory. There have been several generalizations
of the zeta function. Of special interest for our purposes is the polylogarithm function
?(s) :=
? 1
n=0
?(x, s) = Lis(x) := F(x, s) :=
? xn
(1.1)
(1.2)
which extends the zeta function as
?(1, s) = Lis(1) = F(x, 1) = ?(s).
(1.3)
It has been studied extensively by several authors including Lambert, Legendre, Abel,
Kummer, Appell, Lerch, Lindelo?f, Wirtinger, Jonquie`re, Truesdell, and others. It is related
to the Fermi-Dirac and Bose-Einstein integral functions which in turn come from the
Fermi-Dirac and Bose-Einstein statistics for the quantum description of collections of
particles of spins (n + 1/2) and n , respectively. For the asymptotic expansions and other
properties of these functions, we refer to the works in [
2?10
].
We present a series representation of a class of functions and deduce the well-known
series representation and operator forms of the Fermi-Dirac (and Bose-Einstein) integral
and other related functions. The present formulation helps us to find an alternate proof of
the Euler formula for the closed-form representation of the zeta function at even integral
values. Lindelo?f proved the expansion [11, equation (15), page 30]
?(x, s) = Lis(x) = ?(1 ? s)(? log x)s?1 +
?(s ? n)
?
n=0
log (x) n
n!
| log x| < 2? ,
which is useful for numerical evaluation of the function. The function (1.2) is related to
the Fermi-Dirac integral function [4, page 30]
(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
(1.9)
(1.10)
and the Bose-Einstein integral function [4, page 53]
as we have
1 ? t p
Fp(x) := ?(p + 1) 0 et?x + 1 dt (p > ?1),
Bp(x) := ?(p1+ 1) 0? et?txp? 1 dt (p > 0),
Fp?1(x) = ?? ? ex, p = ?Lip ? ex ,
Bp?1(x) = ? ex, p = Lip ex .
Putting p = 0 in (1.5), we find (see also [4, page 20])
Note that the Fermi-Dirac and Bose-Einstein integral functions are also related by the
duplication formula
F0(x) = x + ln 1 + e?x .
Fp(x) = Bp(x) ? 2?pBp(2x),
which is useful in translating the properties of these functions.
2. The Mellin and Weyl transform representations
The Mellin transform of a function ?(t) (0 ? t < ?), if it exists, is defined by (see [12,
page 79])
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
The inversion formula for the Mellin transform is given by [12, page 80]
?M(s) := M[?; s] :=
?
0
ts?1?(t)dt (s = ? + it).
1 c+i?
?(t) = 2?i c?i??M(z)t?zdz.
If ? ? Ll1oc[0, ?) is such that ?(t) = O(t??1 ), t?0+ and ?(t) = O(t??2 ), t??, the integral
?M(s) in (2.1) defines a function in the strip ?1 < ? < ? 2. Moreover, if the function ?(t)
is continuous in [0, ?) and has rapid decay at infinity, the Mellin transform (2.1) will
converge absolutely for ? > 0. In particular, if the integral (2.1) converges uniformly and
absolutely in the strip ?1 < ? < ?2, the function ?M(s) is analytic in the interior of the
strip ?1 < ? < ? 2 [12, page 80].
The Weyl transform of a function ?(t) (0 ? t < ?), if it exists, is defined by [6, page
201]
1 ?
?(s; x) := W ?s ?(t) (x) := ?(s) 0 ts?1?(t + x)dt (? > 0, x > 0).
.
We define ?(0; x) = ?(x) and
?(?s; x) := (?1)n dn
dxn ?(s; x) ,
where n is the smallest integer greater than ?. Then, we have the representation
?(?n; x) := (?1)n dn dxn ?(x)
dxn ?(0; x) = (?1)n dn
(n = 0, 1, 2, . . . ).
Since [6, page 243] W ??W ?? = W ???? = W ??W ??, we have
?(? + ?; x) = W ?? ?(?; t) (x) = ?(1?) 0?t??1?(?; t + x)dt.
3. A class of good functions and applications
To prove our main result, we define a class ? of functions that we call ?good.? A function
? ? Ll1oc[0, ?) is said to be a member of the class ? if
(P.1) ?(0; t) has a power series representation at t = 0 ;
(P.2) the integral in (2.1) is absolutely and uniformly convergent in the strip 0 < ?1 ?
? ? ?2 < 1.
The class of good functions is nonempty, as e?t and (1/(et ? 1) ? 1/t) belong to the class.
We prove our representation formulae here and discuss their applications in the next
sections.
Theorem 3.1. The Weyl transform of a good function, ?, can be represented by
?(s; x) =
?
n=0
?(s ? n; 0) (?x)n
n!
(0 ? ? < 1, x > 0).
Proof. Since ? ? ?, the corresponding function ?(s; x) ? ? must have the Taylor series
expansion about x = 0:
However, we have
is
The proof follows directly from (3.2) and (3.3).
Theorem 3.2. For a good function, ?, the Weyl transform of
? dn xn
?(s; x) = n=0 dxn ?(s; x) x=0 n! .
dn
dxn ?(s; x) x=0 = (?1)n?(s ? n; 0).
?(t) := t?? + ?(t)
?(s; x) = ?(??(??)s) xs?? + ?
?(s ? n; 0) (?x)n
n!
n=0
(0 < ? < ?, x > 0).
cos x + ?2 s =
sin x + ?2 s =
?
n=0
?
n=0
cos ?2 (s ? n)
sin ?2 (s ? n)
(?x)n
n!
,
(?x)n
.
n!
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(4.1)
Proof. Taking the Weyl transform of both sides in (3.4) and using (see [6, equation (7.7),
page 249])
W ?s t?? (x) = ?(? ? s) xs??
?(?)
(0 < ? < ?, x > 0),
we arrive at (3.5).
4. Applications to Fermi-Dirac and Bose-Einstein integral functions
We show that the Fermi-Dirac and Bose-Einstein functions are expressible as the Weyl
transform of a good function and recover their classical series representations by using
the result (3.1) in a simple way. It is to be remarked that the result (3.1) is applicable to a
wider class of functions. For example, cos (t) and sin (t) are good functions having Weyl
transforms cos (x + ?s/2) and sin (x + ?s/2) (0 < ? < 1). An application of (3.1) leads to
the representations
Fs?1(x) =
?
n=0
1 ? 2n?s+1 ?(s ? n) xn
n!
1
?(t) := et + 1 ,
Putting we find that
Putting we find that
Bs?1(?x) = W ?s
with coefficients involving the zeta values.
Proof. Replacing x by ?x in (1.5) and putting p = s ? 1, we obtain the operator form of
the Fermi-Dirac integral
(0 < ? < 1, x > 0),
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.8)
(4.9)
However, we have (see [1, equation (2.7.1), page 23])
Fs?1(?x) = W ?s
1
et + 1 (x) = ?(s; x).
Fs?1(0) = ?(s; 0) = 1 ? 21?s ?(s).
Putting these values in the representation Theorem 3.1 with x in place of ?x, we arrive at
(4.2).
Theorem 4.2. The Bose-Einstein integral function has the series representation
?
n=0
?(s ? n) (?x)n
n!
with coefficients involving the values of the zeta function.
Proof. Replacing x by ?x in (1.6) and putting p = s ? 1, we obtain the operator form of
the integral function
and (see [1, equation (2.7.1), page 23])
Since the function ?(t) ? ?, it follows from the representation Theorem 3.1 that
However, from (3.6), we have
W ?s ?(t) (0) = W ?s
Taking the Weyl transform of both sides in (4.10) and using (4.11)?(4.13), we get
Remark 4.3. The present formulation of the Weyl transform representation of the
FermiDirac integral functions leads to the representation (see (2.6))
1
F?+?(?x) = ?(?) 0?t??1F?(?t ? x)dt (? > 0).
Putting ? = ? ? 1 in (4.15), we get
Similarly, it follows from the operational formulation (4.8) that
1
F2??1(?x) = ?(?) 0?t??1F??1(?t ? x)dt (? > 0).
1
B?+?(?x) = ?(?) 0?t??1B?(?t ? x)dt (? > 0).
The operator representations (4.15) and (4.17) provide a useful relation between the
functions and their transforms.
5. Alternate derivation of Euler?s formula
Euler?s formula relating the Riemann zeta function to the Bernoulli numbers, Bn, is one
of the important results in the theory of the zeta function. The usual derivation is long
and complicated. From the results obtained above for the Fermi-Dirac and Bose-Einstein
integral functions, we obtain the Euler formula more simply. The formula is
?(2n) = (?1)n+1 2((22?n))!2n B2n
(n = 1, 2, 3, ...),
(5.1)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
where the Bernoulli numbers are defined by [2, page 804]
The Euler numbers are defined by
Putting s = 0 in (4.2) and using (1.7), we find that
However, it follows from the Riemann functional equation that
ex
F?1(x) = ex + 1 =
?
n=0
?(?n) 1 ? 2n+1 (xn)!n .
1
?(?n) = ? 2 (2?)?n?1 sin
?n
2
?(n + 1)?(n + 1).
From (5.4) and (5.5), we obtain
2ex
2F?1(x) = ex + 1 = 1 +
?
n=1
2n+1 ? 1 sin
n?
2
?(n + 1)?(n + 1)
(x)n
n!
,
where we have taken
Comparing the coefficients of equal powers of x in (5.3) and (5.6), we get
which can be rewritten to give
lsi?m0 sin s2? ?(s + 1) = 1.
E0 = 1,
E2n = 0 (n = 1, 2, 3, ...),
E2n?1 = (?1)n 4 22n ?(21?)(22nn ? 1)! ?(2n),
?(2n) = (?1)n
(2?)2nE2n?1
4 22n ? 1 (2n ? 1)!
(n = 1, 2, 3, ...).
En =
2 ? 2n+2
n + 1
Bn+1 (n = 1, 2, 3, ...),
(5.2)
(5.3)
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
Now inserting the relation between the Euler and Bernoulli numbers (see [2, page 805])
we obtain (5.1) as desired. The representation of the zeta values at odd integers remains a
challenging task. We hope that the present formulation of the operator representation of
the Fermi integrals may lead to the desired formula.
6. Concluding remarks
Transform techniques are extremely powerful tools for dealing with functions and
constructing solutions of equations. In particular, the Weyl transform, which is at the heart
of the ?fractional calculus,? has been extensively used for various purposes. In this
paper, we have used it to construct the Fermi-Dirac and Bose-Einstein integral functions
from elementary functions. These functions are related to probabilities arising from the
Fermi-Dirac and Bose-Einstein distribution functions which give the quantum
description of collections of identical particles of half-odd integer and integral intrinsic spin,
respectively. Due to their physical significance, these distribution functions have been
extensively studied. Bosons and fermions were regarded as being mutually exclusive with
no fundamental physical quantity described by some function ?between? these two in any
sense. However, it was later realized that there can be ?effective particles,? called anyons,
that are neither fermions nor bosons but something between the two. The process of
obtaining the integral functions by the Weyl transform can be used to develop a candidate
for an anyon integral function.
Our procedure has significant ?spinoffs.? We recover the well-known connections
between the Fermi-Dirac and Bose-Einstein integral functions and with the zeta and
polylogarithm function. Of special interest is an alternative, and extremely elegant, derivation
of the Euler formula relating the Riemann zeta function to the even-integer argument
and the Bernoulli numbers. This demonstrates the significance and power of the Weyl
transform method, as applied here. It also leads us to hope that the Fermi-Dirac integral
function may provide a way of constructing a formula for the odd-integer argument. Zeta
function has remained an open problem for the last 300 years.
Acknowledgments References
The authors are grateful to the King Fahd University of Petroleum and Minerals for
excellent research facilities. This work was carried out under the KFUPM Project IP-2007/21.
M. A. Chaudhry and A. Qadir
M. Aslam Chaudhry: Department of Mathematics and Statistics, King Fahd University of
Petroleum and Minerals, Dhahran 31261, Saudi Arabia
Email address:
Asghar Qadir: Centre for Advanced Mathematics and Physics, National University of Sciences and
Technology, Campus of the College of Electrical and Mechanical Engineering, Peshawar Road,
Rawalpindi, Pakistan
Email address:
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