Integral representations for the Dirichlet L-functions and
their expansions in Meixner-Pollaczek polynomials and
rising factorials ∗
A. Kuznetsov
Dept. of Mathematical Sciences
University of New Brunswick
Saint John, NB
E2L 4L5, Canada
e-mail: akuznets@unbsj.ca
Current version: 10 April, 2007
Abstract
In this article we provide integral representations for the Dirichlet beta and Riemann zeta
functions, which are obtained by combining Mellin transform with the fractional Fourier
transform. As an application of these integral formulas we derive tractable expansions of
these L-functions in the series of Meixner-Pollaczek polynomials and rising factorials.
Keywords: Riemann zeta function, Meixner-Pollaczek polynomials, rising factorials, confluent
hypergeometric function, Mellin transform, fractional Fourier transform
∗
This is the author’s version of the work. It is posted here by permission of Taylor& Francis for personal use,
not for redistribution. The definitive version was published in Integral Transforms and Special Functions, Volume
18, Issue 11, January 2007. doi:10.1080/10652460701450773
1
1
Introduction
In this article we study the relations between Mellin transform and the fractional powers of Fourier
transform and we show how these integral transforms can be used to obtain new results for the
Dirichlet L-functions. To illustrate the ideas let us consider the Dirichlet beta function β(s),
P (−1)n
which for Re(s) > 0 is defined as β(s) =
. This function can be obtained as the Mellin
(2n+1)s
n≥0
pπ
transform of f (x) = sech( 2 x), and the functional equation for β(s) follows from the fact that
f (x) is invariant under cosine transform Fc and the integral kernel xs−1 of the Mellin transform
is also “invariant” under Fc , with s replaced by 1 − s and some multiplication by functions of s
only.
However, more can be said about function f (x): not only it is invariant under Fc , but inm
finitely many fractional powers Fcn f (y) have quite simple form. But since the integral kernel
of the Mellin transform is not invariant under fractional powers of Fc (in fact we obtain a transform with the confluent hypergeometric function as an integral kernel), we obtain new integral
representations for β(s).
These integral representations (Theorems 4.1 and 5.1) provide a simple way of obtaining
tractable expansions of the Dirichlet L-functions in the series of polynomials, such as MeixnerPollaczek polynomials or rising factorials. The partial sums of these expansions also satisfy functional equation and the coefficients have a very simple form of the exponential generating function.
In a companion paper [9] we study in more detail the expansion of the Riemann Ξ-function in
Meixner-Pollaczek polynomials and the zeros of the partial sums of this expansion (see also [2],
[5], [6] and [7]).
This article is organized as follows: in section 2 we provide some background material on the
fractional Fourier transform and its relation to the Mellin transform. Section 3 is devoted to
Mordell integrals, which play the central role in this article since they allow us to compute the
fractional Fourier transform of certain hyperbolic functions. In section 4 we study Dirichlet beta
function, derive integral representation and expansions in Meixner-Pollaczek polynomials and in
rising factorials. Only then in section 5 we study the famous Riemann zeta function ζ(s): it wasn’t
given a priority because in this case analysis is somewhat more complicated by the presence of
the poles.
At last we would like to mention that everywhere in this article we use the following notation
for the rising factorial: (a)n = a(a + 1) . . . (a + n − 1).
2
Fractional Fourier and scaled Mellin transforms
In this section we review some background material on the fractional Fourier sine (cosine) transform and Mellin transform (see [12], [17]) which will be used extensively later. The results are
presented in the framework of the fractional Hankel transform Hνa , which includes both fractional
cosine and sine transforms as the special cases ν = ± 21 .
Let us define A = L2 ([0, ∞), dx) to be Hilbert space of the square integrable functions on
1 2
1
[0, ∞). We choose a complete orthogonal basis φn (x) = Lνn (x2 )e− 2 x xν+ 2 (ν > −1), where Lνn (x)
are Laguerre polynomials (see [8], [4]). The fractional Hankel transform Hνa of order a is a unitary
2
operator on A defined by Hνa φn = e−πina φn (see [17]).
Below we present some of the properties of the fractional Hankel transform which will be used
later:
• Hνa is a unitary operator on A such that Hνa Hνb = Hνa+b and (Hνa )−1 = (Hνa )∗ = Hν−a .
• The integral kernel
1
e− 2 (x
2n!
e−πina Lνn (x2 )Lνn (y 2 ) =
Γ(n
+
ν
+
1)
n≥0
!
πi
e 2 (ν+1)(a−â) 2i cot( πa
xy
2 +y 2 ) √
(x
2 )
e
=
,
xyJν
| sin πa
|
| sin πa
|
2
2
2 +y 2 )
1
(xy)ν+ 2
X
(1)
where â = sign(sin( πa
)) (see [17]).
2
• When ν = − 21 we obtain the fractional cosine transform Fca with the integral kernel
!
r
πi
(a−â)
4
πa
i
2 e
xy
cot( 2 )(x2 +y 2 )
cos
.
1 e2
πa
π | sin
| sin πa
|
|2
2
(2)
2
• When ν =
1
2
we obtain the fractional sine transform Fsa with the integral kernel
!
r
3πi
2 e 4 (a−â) 2i cot( πa
xy
2
2
(x +y )
2 )
sin
.
e
π | sin πa | 21
| sin πa
|
2
2
(3)
Next we define another Hilbert space B = L2 (R, |Γ λ + 2i t |2 dt). Here and everywhere else
in this article λ = ν+1
(note that λ > 0). The scaled Mellin transform is defined as
2
1
s
24−2
(Mf )(s),
(Mλ f )(s) =
Γ λ − 41 + 2s
where (Mf )(s) =
R∞
f (x)xs−1 dx is the classical Mellin transform.
0
The following properties of the scaled Mellin transform will be used later:
(λ)
(λ)
• (Mλ φn )(s) = 2λ−1 (−i)n Pn 2t , where s = 21 + it and Pn 2t are the Meixner-Pollaczek
polynomials (see [8])
• Parseval identity: if g(t) = (Mλ f )(s), s =
1
2
+ it, then ||f ||2A =
1
||g||2B .
2π
• The action of Mλ on Fourier cosine Fc = H− 1 and Fourier sine transforms Fc = H 1 :
2
2
(M 1 Fc f )(s) = (M 1 f )(1 − s), (M 3 Fs f )(s) = (M 3 f )(1 − s).
4
4
4
3
4
(4)
Next we present a result which will be our main tool in the following sections: it describes the
−1
action of the scaled Mellin transform Mλ on the fractional Hankel transform Hν 2 (λ = ν+1
).
2
−1
Proposition 2.1. Assume f (x) ∈ A and let g(t) = (Mλ Hν 2 f )(t). Then g(t) ∈ B, ||f ||2A =
1
||g||2B and g(t) can be represented as
2π
πt
e4
g(t) =
Γ(2λ)
Z∞
1
i 2
x2λ− 2 e− 2 x 1 F1 λ + 2i t, 2λ; ix2 f (x)dx,
0
1
Furthermore, f (x) = (Hν2 M−1
λ g)(x) can be expressed as the following integral
1
i
x2λ− 2 e− 2 x
f (x) =
2πΓ(2λ)
2
Z
1 F1
πt
λ + 2i t, 2λ; ix2 e 4 g(t)|Γ λ + 2i t |2 dt.
R
− 12
Proof: The integral kernel of transformation Mλ Hν
1
1
2
2 e
πi
λ
2
s
24−2
Γ λ − 14 + 2s
Z∞
y
is given by
s−1 − 2i (x2 +y 2 ) √
e
xyJν
√
2xy dy =
0
πt
1
i 2
1
e 4 x2λ− 2 e− 2 x 1 F1 λ + 2i t, 2λ; ix2 .
=
Γ(2λ)
1
Similarly, the integral kernel of the inverse transformation Hν2 M−1
λ is given by
1 1 − πi λ − 1 + s
2 2 e 2 2 4 2 Γ λ − 41 + 2s
2π
Z∞
i
y −s e 2 (x
2 +y 2 )
√
√
xyJν
2xy dy =
0
πt
1
i 2
1
=
e 4 x2λ− 2 e− 2 x 1 F1 λ + 2i t, 2λ; ix2 |Γ λ + 2i t |2 .
2πΓ(2λ)
Both integrals were computed with the help of [4].
3
Mordell integrals
In this section we review several results about function
r Z∞ i 2
2
e 2 τ x cos(xy)
p π dx, Im(τ ) ≥ 0.
h(y, τ ) =
π
cosh
x
2
0
Integrals of this type were used by Riemann to obtain functional equation and asymptotic formula
for zeta function (see [15]). Later these integrals were studied by Ramanujan ([13]) and by Mordell,
4
who analyzed their behavior with respect to modular transformations (see [10], [11]). An extensive
collection of facts about function h(y, τ ) can be found in [16].
In the derivation of the integral representations for Dirichlet L-functions we will need explicit
formulas for h(y, τ ), which can be obtained using the following functional equations (see [16] for
the proof)
√π 2
√
2 πi i
h(y, τ ) + h(y + i 2π, τ ) = √ e 4 − 2τ (y+i 2 ) ,
τ
h(y, τ ) + e−
√
2πy−πiτ
(5)
√ π πiτ
√
h(y + i 2πτ, τ ) = 2e− 2 y− 4 .
(6)
If τ = m
is an irreducible fraction, then by iterating Eq. (5) m times and Eq. (6) n times we
n
√
obtain a system of two linear equations in two variables h1 = h(y, τ ) and h2 = h(y + i 2πm, τ ).
After eliminating h2 from these equations and simplifying the resulting formula for h1 we obtain
pπ
pπ y 1 1 πi − i y2
1
p G 1 ,−
y, τ, n + √ e 4 2τ G 1 ,−
, −τ , m ,
(7)
h(y, τ ) =
2
2τ
2
2
τ
cosh n π2 y
where the quadratic Gauss sum is defined as Ga,± (y, τ, n) =
n−1
P
2 +(n−2k−2a)y
(±1)k e−πiτ (k+a)
. The
k=0
following two integrals can also be expressed in terms of h(y, τ ) and thus computed explicitly for
rational τ :
r Z∞ i 2
πi
πi
i 2
1
2
e 2 τ x sin (xy)
√
dx = − √ e− 4 − 2τ y 1 + Φ e− 4 √y2τ
−
(8)
π
2 τ
e 2πx + 1
0
pπ
pπ y 1 1
1 − πi − i y2
4
2τ
√π
√ π G 1 ,+
−
y, τ, n − √ e
G0,−
, −τ , m
2
2τ
2
τ
en 2 y + (−1)n e−n 2 y
r Z∞ i 2
πi
πi
i 2
2
e 2 τ x sin (xy)
1
√
dx = √ e− 4 − 2τ y 1 + Φ e− 4 √y2τ
+
(9)
π
2 τ
e 2πx − 1
0
"
#
pπ
pπ y 1 1
1 − πi − i y2
√ π G0,+
√π
y, τ, n − √ e 4 2τ G0,+
, −τ , m
+
2
2τ
τ
en 2 y + (−1)n+m e−n 2 y
4
Dirichlet beta function
In this section we illustrate the interplay between Mellin transform and the fractional cosine
transform on the example of the Dirichlet beta function β(s), which for Re(s) > 0 can be defined
P (−1)n
s
as β(s) = L (s, χ4 ) =
. We also define ξ (s, χ4 ) = 2s π − 2 Γ 1+s
β(s) and Ξ (t, χ4 ) =
(2n+1)s
2
n≥0
ξ 12 + it, χ4 .
Our main result in this section is the following integral representation for Ξ(t, χ4 ):
5
Theorem 4.1. For all t ∈ C
πt
Ξ(t, χ4 )e− 4 = 2
Z∞
i
2
e− 2 y 1 F1
1
4
+ 2i t, 12 ; iy 2
+ π8
√
dy.
cosh( πy)
sin
y2
2
(10)
0
pπ x and find that
2
Proof: We start with the function f (x) = sech
1
24
(M 1 f )(s) = √ ξ(s, χ4 ), Re(s) > 0.
4
π
(11)
Note that the functional equation ξ(s, χ4 ) = ξ(1 − s, χ4 ) follows at once from Eq. (4) and the fact
that f (x) is invariant under Fourier cosine transform Fc (see [4]).
1
Now we use Eq. (7) and compute (Fc2 f )(y):
r
1
2
F (y) = (Fc f )(y) =
2
C
π
π
, − 12
4
Z∞
0
2
√
y
π
sin
+
2
8
5
i
cos( 2xy)
2
2
p π dx = 2 4
√
e 2 (x +y )
,
cosh( πy)
cosh
x
2
−1
Next we use the identity f (x) = (Fc 2 F )(x), Eq. (11) and Proposition 2.1 to obtain
1
πt
24
e4
−1
√ ξ(s, χ4 ) = (M 1 Fc 2 F )(s) =
4
π
Γ 21
Z∞
i
2
e− 2 y 1 F1
1
4
+ 2i t, 12 ; iy 2 F (y)dy,
0
which ends the proof.
Corollary 4.2. For all t ∈ C
t
2
(12)
π
√
sin
x
+
√ 8 .
xn = 2π
n!
cosh( 2πx)
(13)
− πt
4
Ξ(t, χ4 )e
=
X
( 14 )
an P n
n≥0
where the generating function for coefficients {an } is
X an
n≥0
Proof: A rigorous proof (and the integral representation for an ) can be obtained by expanding
the confluent hypergeometric function in (10) in Meixner-Pollaczek polynomials (see [8]):
− iy
2
e
∞
X
(−1)n (λ)
Pn (t) y n .
1 F1 (λ + it, 2λ; iy) =
n (2λ)
2
n
n=0
(14)
However we decided to present here a more intuitive argument, which shows why the generating
function for the coefficients an is necessarily the same as the function in the integral representation
(10) (up to a simple change of variables).
6
πt
First we assume that function Ξ(t, χ4 )e 4 lies in the Hilbert space B and we expand it in
P
πt
(1) the orthogonal basis given by Meixner-Pollaczek polynomials: Ξ(t, χ4 )e 4 =
(−1)n an Pn 4 2t .
n≥0
Using the orthogonality relation for the Meixner-Pollaczek polynomials (see [8]) we find that the
coefficients an are given by
√
Z
πt
(−1)n 2n!
( 14 ) t
4
Ξ(t, χ4 )e Pn
|Γ 14 + 2i t |2 dt.
an =
1
2
4πΓ( 2 + n)
R
The generating function for {an } is computed using Eq. (14):
√ −ix Z
X an
πt
2e
1
i
1
xn =
1 F1 4 + 2 t, 2 ; 2ix Ξ(t, χ4 )e 4 |Γ
3
n!
4π 2
n≥0
1
4
+ 2i t |2 dt,
R
and using Proposition 2.1 and the integral representation (10) we find that the above integral
√
sin(x+ π8 )
must be equal to 2π cosh(√2πx)
.
Corollary 4.3. For all t ∈ C
− πt
4
Ξ (t, χ4 ) e
X bn (−i)n
=
n!
n≥0
1
4
+
i
t
2 n
bn i n
+
n!
1
4
−
i
t
2 n
,
(15)
where the generating functions for coefficients {bn } is
X bn
n≥0
n!
n
x =
√
πe
πi
+i x2
4
sin x2 + π8
√
.
cosh( πx)
πt
Proof: Again we start with the integral representation (10) and rewrite it as Ξ (t, χ4 ) e− 4 =
Ψ(t) + Ψ(t), where
Z∞ −iy2
it 1
1
2
e
3πi
1 F1 4 + 2 , 2 ; iy
√
dy.
Ψ(t) = e 8
cosh( πy)
0
Next we use the definition of the confluent hypergeometric function and expand it in the power
series in y (see [4]). Integrating term by term we find that the coefficients bn have the following
integral representation:
bn =
e
3πi
8
(−1)n
1
2 n
Z∞
2
e−iy y 2n
√
dy.
cosh( πy)
(16)
0
√π
One can find using the above formula that for n large |bn | ∼ nα e− 2 n for some α, thus the series
(15) converges for all complex t. The exponential generating function for the coefficients {bn } is
computed using Eq. 7.
7
5
Riemann zeta function
1
s
− 2s
We adopt the following
standard
definitions
(see
[14],[15]):
ξ(s)
=
s(s
−
1)π
ζ(s) and
Γ
2
2
Ξ(t) = ξ 12 + it . Our main result in this section is the following integral representation for Ξ(t):
Theorem 5.1. For all t ∈ C
πt
Ξ(t)e− 4 =
1
cos
2
π
8
π
8
− t sin
+ (1 + 4t2 )
Z∞
i
2
ye− 2 y 1 F1
3
4
+ 2i t, 32 ; iy 2
y2
2
√
e2 πy
sin
+
π
8
+1
dy.
(17)
0
Proof: Define function f (x) =
√
e
1
2πx −1
−
√1 .
2πx
The first step is to find that
1
2 4 ξ(s)
, Re(s) ∈ (0, 1).
(M 3 f )(s) = √
4
π s(s − 1)
(18)
Again, the functional equation ξ(s) = ξ(1 − s) follows from the fact that f is invariant under
Fourier sine transform Fs (see [4]) and Eq. (4).
1
Using Eq. (9) we find the fractional sine transform Fs2 of function f (x):
2


y
π
sin
+
1
2
8
1
1
+
φ(y) − φ(y)  ,
F (y) = (Fs2 f )(y) = 2 4 −2 2√πy
2i
+1
e
y2
(19)
πi 1 − Φ e 4 y . Note that function φ(y) is analytic, and as y → +∞ we
y 2 πi
−i
−
have (see [4]) φ(y) ∼ √1π e 2y 8 + O y12 , thus φ(y) is in the Hilbert space A. Next we find that
πi
where φ(y) = ei 2 + 8
−1
(M 3 Fs 2 φ)(s)
4
πi
πi
i e 4 (1−s)
i e− 4 s
− 12
=√
, (M 3 Fs φ)(s) = √
,
4
π s−1
π s
(20)
and to finish the proof we only need to combine Eqs. (18), (19), (20) and Proposition 2.1.
It is interesting
to note that Eq. (17)
is essentially equivalent to the integral representation
i
h 1
1
1
2
Ξ(t) = 4 + t Υ( 2 + it) + Υ( 2 + it) , where Υ(s) is defined by
s
1 πi
Υ(s) = − e 2 (s−1) 2s−1 π 2 −1 Γ
4
s
2
Z
L
πi
ix2
e 4π
x−s dx.
sinh( x2 )
(21)
and the integral is taken along the line L = e 4 R + πi (see [15]). One can obtain formula (17) by
applying Plancherel theorem for sine transform to the functions inside the integral in Eq. (21).
8
It is also of interest to compare integral representation (17) with the well-known Riemann
formula (see [15]):
Z∞ 1X
2
e−πn x dx =
x n≥1
1
"
#
1−s
2
X Γ s , πn2
,
πn
Γ
√2
= 1 + s(s − 1)
+ √2 1−s .
s
(
πn)
( πn)
n≥1
2ξ(s) = 1 + s(s − 1)
s
x2 + x
1−s
2
(22)
The following proposition shows that Eqs. (22) and (17) are just “extreme” cases (α =
α = 0 correspondingly) of the more general result:
π
2
and
Proposition 5.2. For all α ∈ [0, π]
2ξ(s) = se
i π
( −α)(1−s)
2 2
− 2i ( π2 −α)s
+ (1 − s)e
+ s(s − 1)
"
X Γ
n≥1
π
s
, e−i( 2 −α) πn2
2
√
( πn)s
+
Γ
1−s i( π2 −α)
,e
πn2
2
√
( πn)1−s
#
(23)
and
t
−( π4 − α
2)
Ξ(t)e
1
= cos
2
π
8
−
α
4
− t sin
π
8
−
α
4
2
Z∞
+ (1 + 4t )
i
2
ye− 2 y 1 F1
3
4
+ 2i t, 32 ; iy 2 ϑ(y)dy, (24)
0
√
3πi
αi
i 2 P
iα
+
−
y
2
iα
where ϑ(y) = Re e 8 4 2
exp πin e − 2 πnye 2 .
n≥1
Proof: To derive (23) one should start with the function ψ(y) =
P
2y
eπin
n≥1
= 21 (θ3 (0, y) − 1) and
follow the lines of Riemann’s proof (see [15]), but take Mellin transform along the line y ∈ eiα R+ ,
α ∈ [0, π]. Formula (24) is obtained from (23) with the help of expression for the incomplete
Gamma function as the Laplace transform of the confluent hypergeometric function (see [4]):
π
Z∞
√
Γ 2s , e−i( 2 −α) πn2
iα
−i( π2 −α) 2s
− 2i y 2
2 iα
3
i
3
2
2
√
ye
F
dy.
=
4ie
πnye
+
t,
;
iy
exp
πin
e
−
2
1 1 4
2 2
( πn)s
0
Next we derive an expansion of the Riemann Xi function in the Meixner-Pollaczek polynomials
(see [9] for the detailed analysis of the coefficients of this expansion and zeros of its partial sums).
Corollary 5.3. For all t ∈ C
Ξ(t)e
− πt
4
1
= cos
2
π
8
− t sin
π
8
2
+ (1 + 4t )
X
n≥0
( 34 )
an P n
t
,
2
(25)
where the generating function for coefficients {an } is
r √
1 πi √ πi √ X an
πi
πi
1 π
n
ix+
−
−ix−
π
8
x =
− sin x + 8 tanh
2πx +
Φ e4 x e 8 −Φ e 4 x e
.
n!
4
x
2i
n≥0
9
Proof: Again we start with Eq. (17), expand the confluent hypergeometric function in the series
of Meixner-Pollaczek polynomials (see Eq. (14)) and integrate term by term to find that the
coefficients are given by
2
∞
y
π
Z
sin 2 + 8
(−1)n
√
an =
y 2n+1 dy.
πy
2
(2n + 1)!!
+1
e
0
The exponential generating function for {an } is computed using Eq. (8).
πt
Following the lines of the proof of Corollary 4.3 we obtain the following expansion of Ξ(t)e− 4
in rising factorials:
Corollary 5.4. For all t ∈ C
− πt
4
Ξ(t)e
1
= cos
2
π
8
− t sin
π
8
X bn (−i)n
+ (1 + 4t )
n!
n≥0
2
3
4
+
i
t
2 n
bn in
+
n!
3
4
−
i
t
2 n
,
where the generating function for the coefficients {bn } is
n≥0
3πi
e− 8
xn =
n!
16
X bn
r √
π ix πi √ 1 − eix cosh( πx)
4
√
e Φ e
x +
.
x
sinh( πx)
Acknowledgment:
The first version of the manuscript was completed while the author was a Postdoctoral Fellow
at the Department of Mathematics and Statistics, McMaster University. The author would like
to thank an anonymous referee for many helpful comments.
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