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Acta Math. Hungar.
DOI: 0
INTEGRAL REPRESENTATIONS AND
SUMMATIONS OF THE MODIFIED STRUVE
FUNCTION
2,∗
´
´ BARICZ1 and T. K. POGANY
A.
1
Department of Economics, Babe¸s–Bolyai University, Cluj-Napoca 400591, Romania
e-mail: bariczocsi@yahoo.com
2
Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia
e-mail: poganj@pfri.hr
(Received October 18, 2012; revised November 12, 2012; accepted November 12, 2012)
Abstract. It is known that the Struve function Hν and the modiﬁed Struve
function Lν are closely connected to the Bessel function of the ﬁrst kind Jν and
to the modiﬁed Bessel function of the ﬁrst kind Iν and possess representations
through higher transcendental functions like generalized hypergeometric 1 F2 and
the Meijer G function. Also, the NIST project and Wolfram formula collection
contain a set of Kapteyn type series expansions for Lν (x). In this paper ﬁrstly, we
obtain various another type integral representation formulae for Lν (x) using the
technique developed by D. Jankov and the authors. Secondly, we present some
summation results for diﬀerent kind of Neumann, Kapteyn and Schl¨
omilch series
built by Iν (x) and Lν (x) which are connected by a Sonin–Gubler formula, and by
the associated modiﬁed Struve diﬀerential equation. Finally, solving a Fredholm
type convolutional integral equation of the ﬁrst kind, Bromwich–Wagner line integral expressions are derived for the Bessel function of ﬁrst kind Jν and for an
associated generalized Schl¨
omilch series.
1. Introduction
The Bessel and modiﬁed Bessel function of ﬁrst kind, the Struve and
modiﬁed Struve function possess power series representation of the form [61]:
Jν (z) =
∑ (−1)n ( x2 )2n+ν
,
Γ(n + ν + 1)n!
n =0
∗ Corresponding
Iν (z) =
∑
n=0
( x2 )
2n+ν
Γ(n + ν + 1)n!
author.
Key words and phrases: modiﬁed Struve function, Bessel function and modiﬁed Bessel function of the ﬁrst kind, Neumann, Kapteyn and Schl¨
omilch series of modiﬁed Bessel and Struve
functions, Dirichlet series, Cahen formula, generalized hypergeometric function, Struve diﬀerential
equation.
Mathematics Subject Classification: primary 33C10, 33E20, 40H05, secondary 30B50, 40C10,
65B10.
c 0 Akad´
0236-5294/$ 20.00 ⃝
emiai Kiad´
o, Budapest
2
Hν (z) =
´ BARICZ and T. K. POGANY
´
A.
∑
n =0
2n+ν+1
(−1)n ( x2 )
Γ(n +
3
2
)Γ(n + ν + 23 )
,
∑
2n+ν+1
( x2 )
Lν (z) =
,
Γ(n + 23 )Γ(n + ν + 32 )
n=0
where ν ∈ R and z ∈ C. Struve [54] introduced the Hν function as the series
solution of a nonhomogeneous second order Bessel type diﬀerential equation (which carries his name). However, the modiﬁed Struve function Lν
appeared in the mathematical literature due to Nicholson [38, p. 218]. Applications of Struve functions are manyfold and include among others optical investigations [58, pp. 392–395]; general expression of the power carried
by a transverse magnetic or electric beam, is given in terms of Ln+ 1 [2];
2
triplet phase shifts of the scattering by the singular nucleon-nucleon potentials ∝ exp (−x)/xn [20]; leakage inductance in transformer windings [25];
boundary element solutions of the two-dimensional multi-energy-group neutron diﬀusion equation which governs the neutronic phenomena in nuclear
reactors [26]; eﬀective isotropic potential for a pair of dipoles [34]; perturbation approximations of lee-waves in a stratiﬁed ﬂow [35]; quantum-statistical
distribution functions of a hard-sphere system [40]; scattering of plane waves
by circular cylinders for the general case of oblique incidence and for both
real and complex values of particle refractive index [52]; aerodynamic sensitivities for subsonic, sonic, and supersonic unsteady, non-planar liftingsurface theory [14]; stress concentration around broken ﬁlaments [19] and
lift and downwash distributions of oscillating wings in subsonic and supersonic ﬂow [59,60].
Series of Bessel and/or Struve functions in which summation indices appear in the order of the considered function and/or twist arguments of the
constituting functions, can be uniﬁed in a double lacunary form:
∑
αn Bℓ1 (n) (ℓ2 (n)z),
Bℓ1 ,ℓ2 (z) :=
n =1
where x 7→ ℓj (x) = µj + aj x, j ∈ {1, 2}, x ∈ {0, 1, . . . }, z ∈ C and Bν is one
of the functions Jν , Iν , Hν and Lν . The classical theory of the Fourier–
Bessel series of the ﬁrst type is based on the case when Bν = Jν , see the
celebrated monograph by Watson [61]. However, varying the coeﬃcients of
ℓ1 and ℓ2 , we get three diﬀerent cases which have not only deep roles in
describing physical models and have physical interpretations in numerous
topics of natural sciences and technology, but are also of deep mathematical
interest, like e.g. zero function series [61]. Hence we diﬀer: Neumann series
(when a1 ̸= 0, a2 = 0), Kapteyn series (when a1 · a2 ̸= 0) and Schl¨omilch series (when a1 = 0, a2 ̸= 0). Here, all three series are of the ﬁrst type (the
series’ terms contain only one constituting function Bν ); the second type
series contain product terms of two (or more) members – not necessarily different ones – from Jν , Iν , Hν and Lν . We also point out that the Neumann
Acta Mathematica Hungarica
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MODIFIED STRUVE FUNCTION
series (of the ﬁrst type) of Bessel function of the second kind Yν , modiﬁed
Bessel function of the second kind Kν and Hankel functions (Bessel func(1)
(2)
tions of the third kind) Hν , Hν have been studied by Baricz, Jankov and
Pog´
any [5], while Neumann series of the second type were considered by
Baricz and Pog´
any in somewhat diﬀerent purposes in [6,7]; see also [30]. An
important role has throughout this paper the Sonin–Gubler formula which
connects modiﬁed Bessel function of the ﬁrst kind Iν , modiﬁed Struve function Lν and a deﬁnite integral of the Bessel function of the ﬁrst kind Jν [22,
p. 424] (actually a special case of a Sonin-formula [61, p. 434]):
∫ ∞
)
Jν (ax) dx
π (
(1.1)
=
I
(an)
−
L
(an)
,
ν
ν
x2 + n2 xν
2nν+1
0
where ℜ(ν) > − 12 , a > 0 and ℜ(n) > 0; see [61, p. 426] for the historical
background of (1.1).
Thus, under extended Neumann series (of Bessel Jν see [61]) we mean
the following
∑
NB
(x)
:=
βn Bµn+η (ax),
µ,η
n =1
where Bν is one of the functions Iν and Lν . Integral representation discussions began very recently with the introductory article by Pog´any and
S¨
uli [45], which gives an exhaustive references list concerning physical applications too; see also [5]. In Section 2 we will concentrate on the Neumann
series
∑
(1.2)
Nµ,η (x) :=
βn Iµn+η (ax).
n =1
Secondly, Kapteyn series of the ﬁrst type [31,32,39] are of the form
∑
(
)
KB
αn Bρ+µn (σ + νn)z ;
(z)
:=
ν,µ
n=1
more details about Kapteyn and Kapteyn-type series for Bessel function can
be found also in [4,6,15,55] and the references therein. Here we will consider
speciﬁc Kapteyn-type series of the following form:
∑ αn (
)
(1.3)
Kαν,µ (x) :=
I
(xn)
−
L
(xn)
;
νn
νn
nµ
n =1
this series appear as an auxiliary expression in the fourth section of the article. Thanks to Sonin–Gubler formula (1.1) we give an alternative proof
Acta Mathematica Hungarica
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´ BARICZ and T. K. POGANY
´
A.
for the integral representation of Kαν,µ (x), see Section 3. Thirdly, under Schl¨
omilch series [50, pp. 155–158] (Schl¨omilch considered only cases
µ ∈ {0, 1}), we understand the functions series
∑
(
)
(1.4)
SB
αn Bµ (ν + n)z .
µ,ν (z) :=
n =1
Integral representation are recently obtained for this series in [29], summations are given in [57]. Our attention is focused currently on
∑ αn (
)
SI,L
(1.5)
Iν (xn) − Lν (xn) .
µ,ν (z) :=
µ
n
n =1
The next generalization is suggested by the theory of Fourier series, and the
functions which naturally come under consideration instead of the classical
sine and cosine, are the Bessel functions of the ﬁrst kind and Struve’s functions. The next type series considered here we call generalized Schl¨omilch
series [61, p. 622], [27, p. 1803]
( x )−ν ∑ a J (nx) + b H (nx)
a0
n ν
n ν
+
.
2Γ(ν + 1)
2
nν
n=1
For further subsequent generalizations consult e.g. Bondarenko’s recent article [9] and the references therein and Miller’s multidimensional expansion [36]. A set of summation formulae of Schl¨omilch series for Bessel function of the ﬁrst kind can be found in the literature, such as the Nielsen
formula [61, p. 636]; further, we have [56, p. 65], also consult [21,43,48,57,
62,63]. Similar summations, for Schl¨
omilch series of Struve function, have
been given by Miller [37], consult [57] too.
Further, we are interested in a speciﬁc variant of generalized Schl¨omilch
series in which Jν , Hν are exchanged by Iν and Lν respectively, when an , bn
are of the form a0 = 0, an = 2−ν nν−µ xν = −bn , µ = ν > 0, which results in
(1.6)
TI,L
ν,µ (x) :=
∑ Iν (nx) − Lν (nx)
n =1
nµ
.
n−1
e I,L
Its alternating variant T
an 7→ an , where
ν,µ (x) we perform setting (−1)
n ∈ {0, 1, . . . }:
(1.7)
e I,L (x) :=
T
ν,µ
∑ (−1)n−1 (
n =1
Acta Mathematica Hungarica
nµ
)
Iν (nx) − Lν (nx) .
5
MODIFIED STRUVE FUNCTION
Summations of these series are one of tools in obtaining explicit expressions
for integrals containing Butzer–Flocke–Hauss complete Omega-function
e
[33,47].
Ω(x) [10–12] and Mathieu series S(x), S(x)
Rayleigh [49] has showed that series SB
0,ν (z) play important roles in
physics, because they are useful in investigation of a periodic transverse vibrations uniformly distributed in direction through the two dimensions of
the membrane. Also, Schl¨
omilch series present various features of purely
mathematical interest and it is remarkable that a null-function can be represented by such series in which the coeﬃcients are not all zero [61, p. 622].
Summation results in form of a double deﬁnite integral representation for
SJµ,ν (z), achieved via Kapteyn-series, have been recently derived in [28].
Finally, we mention that except the Sonin–Gubler formula (1.1) another
main tool we refer to is the Cahen formula on the Laplace integral representation of Dirichlet series. Namely, the Dirichlet series
∑
Da (r) =
an e−rλn ,
n =1
where ℜ(r) > 0, having positive monotone increasing divergent to inﬁnity sequence (λn )n=1 , possesses Cahen’s integral representation formula [13, p. 97]
∫
(1.8)
∞
Da (r) = r
−rt
e
0
∑
∫
∞ ∫ [λ−1 (t)]
du a(u) dt du,
an dt = r
0
n: λn 5t
0
d
where dx := 1 + {x} dx
. Here, [x] and {x} = x − [x] denote the integer and
fractional part of x ∈ R, respectively. Indeed, the so-called counting sum
∑
Aa (t) =
an
n: λn 5t
we ﬁnd by the Euler–Maclaurin summation formula, following the procedure
developed by the second author [44], see also [47]. Namely
Aa (t) =
−1
(t)]
[λ∑
n=1
∫ [λ−1 (t)]
du a(u) du,
an =
0
since λ : R+ 7→ R+ is monotone, there exists unique inverse λ−1 for the function λ : R+ 7→ R+ , λ|N = (λn ).
Acta Mathematica Hungarica
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´ BARICZ and T. K. POGANY
´
A.
2. Lν as a Neumann series of modiﬁed Bessel I functions
Let us observe the well-known formulae [42, Eqs. 11.4.18–19–20]

∑ (2n + ν + 1)Γ(n + ν + 1)
4


J2n+ν+1 (z)
√

1

n!(2n + 1)(2n + 2ν + 1)
π Γ(ν + 2 )


n =0



√

z n

 z ∑ (2)
Jn+ν+ 1 (z)
Hν (z) =
2
2π
n!(n + 21 )

n
=
0




1

n

z ν+

( z2 )
 (2) 2 ∑

Jn+ 1 (z),


2
Γ ν + 1
n! n + ν + 1
(
2
) n =0 (
2
)
where the ﬁrst formula is valid for −ν ̸∈ N. So, having in mind that Lν (z) =
−i1−ν Hν (iz) and Jν (iz) = iν Iν (z), we immediately conclude that
(2.1)

∑ (−1)n (2n + ν + 1)Γ(n + ν + 1)
4


I2n+ν+1 (z)
√


n!(2n + 1)(2n + 2ν + 1)
π Γ(ν + 12 )


n =0




√

z n

 z ∑ (− 2 )
In+ν+ 1 (z)
Lν (z) =
2
2π
n!(n + 21 )

n
=
0




1

n

 ( z2 )ν+ 2 ∑
( − z2 )



In+ 1 (z).

2
 Γ(ν + 1 )
n!(n + ν + 1 )
2
n =0
2
However, all three series expansions we recognize as Neumann series built by
modiﬁed Bessel functions of the ﬁrst kind. This kind of series have been intensively studied very recently by the authors and D. Jankov in [5]. Exploiting the appropriate ﬁndings of that article, we give new integral expressions
for the modiﬁed Struve function Lν .
First let us modestly generalize [5, Theorem 2.1] which concerns N1,ν (x),
to integral expression for Nµ,η deﬁned by (1.2), following the same procedure
as in [5].
Theorem 1. Let β ∈ C1 (R+ ), β|N = {βn }n∈N , µ > 0 and assume that
1
(2.2)
Acta Mathematica Hungarica
|βn | µn
µ
lim
< .
n→∞
n
e
7
MODIFIED STRUVE FUNCTION
Then, for µ, η such that min {η + 23 , µ + η + 1} > 0 and
(
( (
)−1 ))
µ
1/n
−µ
x ∈ 0, 2 min 1, (e/µ) lim sup n |βn |
:= Iβ ,
n→∞
we have the integral representation
( (
)
)
∫ ∞ ∫ [u]
∂
1
(2.3)
Nµ,η (x) = −
Γ µu + η +
Iµu+η (x)
∂u
2
1
0
(
)
β(s)
× ds
du ds.
Γ(µs + η + 12 )
Proof. The proof is a copy of the proving procedure delivered for [5,
Theorem 2.1]. The only exception is to reﬁne the convergence condition
upon Nµ,η (x). By the bound [3, p. 583]
Iν (x) <
( x2 )
ν
Γ(ν + 1)
x2
e 4(ν+1) ,
where x > 0 and ν + 1 > 0, we have
∑
( )µ+η
x2
|βn |
Nµ,η (x) < x
e 4(µ+η+1)
,
2
Γ(µn + η + 1)
n =1
so, the absolute convergence of the right hand side series suﬃces for the
ﬁniteness of Nµ,η (x) on Iβ . However, condition (2.2) ensures the absolute
convergence by the Cauchy convergence criterion.
The remaining part of the proof mimicks the one performed for [5, Theorem 2.1], having in mind that µ = 1 reduces Theorem 2 to the ancestor
result [5, Theorem 2.1]. Theorem 2. If ν > 0 and x ∈ (0, 2), then we have the integral representation
(2.4)
∫
∞ ∫ [u]
=
1
0
2Γ(ν + 2)
Lν (x) − √
Iν+1 (x)
π Γ(ν + 32 )
( (
)
) (
3
β(s)
∂
Γ 2u + ν +
I2u+ν+1 (x) ds
∂u
2
Γ(2s + ν +
)
3
2
)
du ds,
where
eiπs (2s + ν + 1)Γ(s + ν + 1)
β(s) = − √
π Γ(ν + 12 )Γ(s + 1)(s + 12 )(s + ν +
1
2
)
.
Acta Mathematica Hungarica
8
´ BARICZ and T. K. POGANY
´
A.
Proof. Consider the ﬁrst Neumann sum expansion of Lν (x) in (2.1),
that is
Lν (x) = √
4
π Γ(ν +
∑ (−1)n (2n + ν + 1)Γ(n + ν + 1)
1
2
) n=0
n!(2n + 1)(2n + 2ν + 1)
I2n+ν+1 (x)
2Γ(ν + 2)
=√
Iν+1 (x)
π Γ(ν + 32 )
4
−√
π Γ(ν +
∑ (−1)n−1 (2n + ν + 1)Γ(n + ν + 1)
1
2
) n =1
n!(2n + 1)(2n + 2ν + 1)
I2n+ν+1 (x).
Observe that
2Γ(ν + 2)
Lν (x) = √
Iν+1 (x) − N2,ν+1 (x)
π Γ(ν + 32 )
in which we specify
βn = √
(−1)n−1 (2n + ν + 1)Γ(n + ν + 1)
π Γ(ν + 12 )Γ(n + 1)(n + 12 )(n + ν +
1
2
)
.
Since
ν−2
β(s) ∼ √ 2s
π Γ(ν +
1
2
)
,
s → ∞,
we deduce (by means of Theorem 1) that (2.4) is valid for x ∈ Iβ = (0, 2).
Theorem 3. For ν + 2 > 0 and x ∈ (0, 2) we have the integral representation
√
2x
(2.5)
Lν (x) −
I 1 (x)
π ν+ 2
∫
(2.6)
∞ ∫ [u]
=
1
0
)
) (
∂ (
β(s)
du ds,
Γ(u + ν + 1)Iu+ν+ 1 (x) ds
2
∂u
Γ(s + ν + 1)
where
√
β(s) = −
Acta Mathematica Hungarica
x s
eiπs ( 2 )
x
2π Γ(s + 1)(s +
1
2
)
.
9
MODIFIED STRUVE FUNCTION
Proof. Let us observe now the second Neumann sum expansion of
Lν (x) in (2.1):
√
x n
x ∑ (− 2 )
Lν (x) =
In+ν+ 1 (x)
2
2π
n!(n + 12 )
n =0
√
=
2x
I 1 (x) −
π ν+ 2
In other words,
√
√
Lν (x) =
n−1 x n
(2)
x ∑ (−1)
In+ν+ 1 (x).
2
2π
n!(n + 21 )
n=1
2x
I 1 (x) − N1,ν+ 1 (x)
2
π ν+ 2
in which we specify
√
s
eiπs ( x2 )
x
β(s) = −
2π Γ(s + 1)(s +
1
2
)
.
The convergence condition (2.2) reduces to the behavior of the auxiliary
series
√
)
(
1
∑
|x|
|βn |
2x
2
,
∼
F
1
2
3
1
π
Γ(n + ν + 12 )
2, ν + 2 2
n =0
which is convergent for all bounded x ∈ C, unconditionally upon ν. Here
1 F2 denotes the hypergeometric function deﬁned by series [1, p. 62]
)
(
∑
(a)n
zn
a z
=
F
.
1 2
b1 , b2 (b1 )n (b2 )n n!
n=0
However, for ν > −2 we have the integral expression [61, p. 79]
∫ 1
(
) ν− 1
21−ν z ν
2
(2.7)
cosh (zt) dt,
Iν (z) = √
1 − t2
1
π Γ(ν + 2 ) 0
where z ∈ C and ℜ(ν) > − 12 . This was used in the proof of the ancestor
result (2.3), see [5, Theorem 2.1]. Now, we apply Theorem 1 and conclude
that (2.5) is valid for x ∈ Iβ = (0, 2). The third formula in (2.1) reduces to the case N1, 1 (x). However, we
2
shall omit the proof, since the slightly repeating derivation procedure used
for (2.5) directly gets the desired integral expression.
Acta Mathematica Hungarica
10
´ BARICZ and T. K. POGANY
´
A.
Theorem 4. If ν + 2 > 0 and x ∈ (0, 2), then we have the integral representation
Lν (x) −
∫
∞ ∫ [u]
=
1
0
xν sinh x
√
2ν π Γ(ν +
∂
(Γ(u + 1)Iu+ 12 (x))ds
∂u
3
2
(
)
β(s)
Γ(s + 1)
)
du ds,
where
ν+ 12
s
eiπs ( x2 )
(x)
β(s) = − 2
.
Γ(ν + 21 ) Γ(s + 1)(s + ν + 12 )
Now, applying (2.7) we derive another integral expression for Lν (x) in
terms of hypergeometric functions in the integrand.
Theorem 5. Let ν > − 12 . Then for x > 0 we have
xν+1 Γ(ν + 2)
Lν (x) = √ 2ν− 1
3
ν
3
ν
2 Γ(ν +
π2
2 )Γ( 2 + 4 )Γ( 2 +
5
4
(
× 4 F5
∫
3
2,
1 ν+3
1
2, 2 , ν + 2, ν + 1
ν+1 ν
3 ν
5
3
2 , 2 + 4, 2 + 4, ν + 4
)
1(
1 − t2
) ν+ 1
2
cosh (xt)
0
)2
x2 (
−
1 − t2
16
)
dt,
where
(
4 F5
4
∏
(a )
)
∑ j=1 j n z n
a1 , a2 , a3 , a4 z =
5
b1 , b2 , b3 , b4 , b5 ∏
n!
n=0
(bj )n
j=1
stands for the generalized hypergeometric function with four upper and ﬁve
lower parameters.
Proof. Consider the ﬁrst Bessel function series expansion for Lν (x)
given in (2.1). Applying mutatis mutandis the integral representation
formula (2.7), the Pochhammer symbol technique, the familiar formula
(A)n (n + A) = A(A + 1)n , n ∈ {0, 1, . . . }, and the duplication formula
(
)
22z−1
1
√
Γ(2z) =
Γ(z)Γ z +
2
π
Acta Mathematica Hungarica
11
MODIFIED STRUVE FUNCTION
to the summands, we get the chain of equivalent legitimate transformations:
8
Lν (x) = √
π Γ(ν +
2( x2 )
∑ (−1)n (2n + ν + 1)Γ(n + ν + 1)
1
2
) n =0
2n+ν+1
∫
×√
π Γ(2n + ν +
=
×
4( x2 )
(n
n =0
ν+1
=√
4( x2 )
π (ν +
×
1
2
1
2
)
)
1(
1 − t2
1(
1 − t2
2
cosh (xt) dt
) ν+ 1
cosh (xt)
2
0
2
(
)2
)Γ(n + ν + 1)[ − x4 1 − t2 ]
)(n + ν + 12 )Γ(2n + ν + 23 )n!
ν+1
2
+ 12
∫
(ν + 1)Γ(ν + 1)
1
2
)Γ(ν + )Γ(
ν
2
+
3
4
)Γ(
ν
2
∑ ( 21 )
n=0
) 2n+ν+ 1
0
∫
ν+1
πΓ(ν +
∑ (n +
3
2
n!(2n + 1)(2n + 2ν + 1)
+
5
4
)
1(
dt
) ν+ 1
2
cosh (xt)
0
(
2
ν+3
ν + 21 n (ν + 1)n − x16
2
n
n
3
ν+1
ν
+ 43 n ν2 + 54 n ν
2 n
2
n 2
( ) (
)
( ) ( ) (
) (
1 − t2
n
[
) (
1 − t2
+
)2
3
4 n n!
)
n
]
dt,
which proves the assertion. By virtue of similar manipulations presented above, we conclude the following results.
Theorem 6. Let ν > − 12 and x > 0. Then there holds
Lν (x)

(
)
∫ 1
x2 (
1
ν+1

(
)ν
)
x

2
2

1−t
cosh (xt)1 F2 3 2
1−t
dt

−

 2ν−1 πΓ(ν + 1) 0
4
2, ν + 1
=
(
)
∫ 1

1 ν+1
2(
)

x
x
ν
+

2 −

cosh (xt)1 F2
1 − t2
dt.

3  √π 2ν+ 12 Γ ν + 1
4
1,
ν
+
( 2) 0
2
The proof of Theorem 6 follows from the same proving procedure as the
previous theorem but now considering the second and third series expansion
results in (2.1), so we shall omit the proofs of these integral representations.
Acta Mathematica Hungarica
12
´ BARICZ and T. K. POGANY
´
A.
3. Integrals containing Ω(x)-function and Mathieu series via
L
summation of TI,
ν (x)
By virtue of the Sonin–Gubler formula (1.1) we establish the convergence
e I,L
conditions for the generalized Schl¨
omilch series TI,L
ν,µ (x) and Tν,µ (x). As for
n enough large we have
∫
( ν−1 )
2nν−1 ∞
J (ax) dx
( ν
)
(3.1)
Iν (an) − Lν (an) =
=
O
n
,
π
1 + n−2 x2 xν
0
we immediately conclude that the following equiconvergences hold true
e I,L (x) ∼ η(µ − ν + 1),
T
ν,µ
TI,L
ν,µ (x) ∼ ζ(µ − ν + 1),
e I,L
that is, TI,L
ν,µ (x) converges for µ > ν > 0, while Tν,µ (x) converges for µ + 1 >
e I,L
ν > 0. On the other hand, we connect T
ν,ν (x) and the Butzer–Flocke–Hauss
(BFL) complete Omega function [10, Deﬁnition 7.1]
∫
1
2
sinh (wu) cot (πu) du,
Ω(w) = 2
w ∈ C.
0+
By the Hilbert transform terminology, Ω(w) is the Hilbert transform
H (e−wx )1 (0) at 0 of the 1-periodic function (e−wx )1 deﬁned by the periodic continuation of the following exponential function [10, p. 67]: e−xw ,
x ∈ [ − 12 , 12 ), w ∈ C, that is,
−xw
H (e
∫
)1 (0) := P.V.
1
2
−
1
2
ewu cot (πu) du ≡ Ω(w)
where the integral is to be understood in the sense of Cauchy’s Principal
Value at zero, see e.g. [12,46].
On the other hand by diﬀerentiating once (1.1) with respect to n we get
´
a tool to obtain Mathieu series S(x) (introduced by Emile
Leonard Mathe
ieu [33]) and its alternating variant S(x) (introduced by Pog´any, Srivastava
and Tomovski [47]), which are deﬁned as follows
S(x) =
∑
n =1
(
2n
x 2 + n2
)2 ,
e
S(x)
=
∑ 2(−1)n−1 n
(
)2 .
x2 + n2
n =1
Closed integral expression for S(r) was considered by Emersleben [18] and
subsequently by Elbert [17], while for Seµ (x) integral representation has been
Acta Mathematica Hungarica
13
MODIFIED STRUVE FUNCTION
given by Pog´
any, Srivastava and Tomovski [47]:
∫
1 ∞ t sin (xt)
(3.2)
S(x) =
dt,
x 0
et − 1
∫
1 ∞ t sin (xt)
e
(3.3)
S(x) =
dt.
x 0
et + 1
Another kind of integral expressions for the underlying Mathieu series can
be found in [47].
Theorem 7. Assume that ℜ(ν) > 0 and a > 0. Then we have
∫ [et ]
∫ ∞
∫ ∞
Jν (ax)Ω(2πx)
−νt
du (eiπu (Lν (au) − Iν (au))) dt du.
dx = ν
e
xν sinh (πx)
0
0
0
Proof. When we multiply (1.1) by (−1)n−1 n and sum up all three series with respect to n ∈ N, the following partial-fraction representation of
the Omega function [10, Theorem 1.3]
∞
∑ 2(−1)n−1 n
πΩ(2πw)
=
.
sinh (πw) n=1 n2 + w2
immediately give
∫ ∞
∑
(
)
Jν (ax)Ω(2πx)
e I,L (a).
dx =
(−1)n−1 n−ν Iν (an) − Lν (an) = T
ν,ν
ν
x
sinh
(πx)
0
n =1
We recognize the right hand side sums as Dirichlet series of Iν and Lν , respectively. Being
∑
∑
eiπ(n−1) Iν (an)e−ν ln n ,
ℜ(ν) > 0,
(−1)n−1 n−ν Iν (an) =
n =1
n =1
we get
∑
n=1
n−1 −ν
(−1)
n
∫
∞
Iν (an) = ν
e−νt
0
∑
eiπ(n−1) Iν (an) dt.
n: ln n5t
So, making use of the Euler–Maclaurin summation (1.8) to the Cahen’s formula we deduce
∫ ∞ ∫ [et ]
∑
(
)
n−1 −ν
(−1)
n Iν (an) = −ν
e−νt du eiπu Iν (au) dt du;
n =1
0
0
Acta Mathematica Hungarica
14
´ BARICZ and T. K. POGANY
´
A.
and repeating the procedure to the second Dirichlet series containing Lν (an),
the proof is complete. The next result concerns a hypergeometric integral, which we integrate
by means of Schl¨
omilch series of modiﬁed Bessel and modiﬁed Struve functions.
Theorem 8. Let ℜ(ν) > 0 and a > 0. Then we have
(1
√ ν+n2 ∫ 1
∫ ∞
t2
dx
πa
2,
Jν (ax)S(x) ν = ν+1
F
1
at − 1 2 1
x
e
2
Γ(ν
+
)
0
0
2
1
2 −
3
2
)
ν 2
dt
t
[
]
πaν Li2 (e−a ) + aLi1 (e−a )
,
+
2ν+1 Γ(ν + 1)
where Liα (z) stands for the dilogarithm function.
Proof. Diﬀerentiating (1.1) with respect to n, we get
∫ ∞
2nJν (ax) dx
(
)2 ν
0
x2 + n2 x
=
)
)
aπ (
π(ν + 1) (
Iν (an) − Lν (an) − ν+1 Iν′ (an) − L′ν (an) .
ν+2
2n
2n
Summing up this relation with respect to positive integers n ∈ N, we get
∫ ∞
)
π(ν + 1) ∑ −ν−2 (
dx
n
Iν (an) − Lν (an)
κν (a) :=
Jν (ax)S(x) ν =
x
2
0
n =1
−
)
aπ ∑ −ν−1 ( ′
aπ d I,L
π(ν + 1) I,L
n
Iν (an) − L′ν (an) =
Tν,ν+2 (a) −
T
(a).
2
2
2 da ν,ν+2
n =1
Now, by the Emersleben–Elbert formula (3.2) we conclude that
)
(∫ ∞
∫ ∞
∫ ∞
dx
t
Jν (ax) sin (xt)
κν (a) =
Jν (ax)S(x) ν =
dx dt.
x
et − 1
xν+1
0
0
0
Expressing the sine via J 1 , we get that the innermost integral equals
2
∫
∞
(3.4)
0
Jν (ax) sin (xt)
dx =
xν+1
Acta Mathematica Hungarica
√
πt
2
∫
∞
Jν (ax)J 1 (tx)
2
0
x
ν+ 12
dx.
15
MODIFIED STRUVE FUNCTION
Now, we shall apply the Weber–Sonin–Schafheitlin formula [61, §13.41] for
λ = ν + 21 , which reduces to
∫ ∞
1
Jν (ax)J 1 (tx)x−ν− 2 dx
0
√

aν π


,

1√

 2ν+ 2 t Γ(ν + 1)
(
=
√
1
ν−1 t

a

2,


F
2 1
 ν+ 12
2
Γ(ν + 12 )
2
if 0 < a 5 t
1
2 −
3
2
)
ν t2
2 , if 0 < t < a.
a
Accordingly, (3.4) becomes
2)
(1 1
√ ν−1 ∫ a
t
πa
t2
,
−
ν
2 2
κν (a) = ν+1
2 F1
3
a2 dt
1
t
2 Γ(ν + 2 ) 0 e − 1
2
∫ ∞
πaν
t
+ ν+1
dt
t
2 Γ(ν + 1) a e − 1
)
(1 1
√ ν+2 ∫ 1
πa
t2
, 2 − ν 2
2
dt
= ν+1
2 F1
3
t
2 Γ(ν + 12 ) 0 eat − 1
2
∫ ∞
πaν
t+a
+ ν+1
dt
2 Γ(ν + 1) 0 et+a − 1
)
(1 1
√ ν+2 ∫ 1
πa
t2
, 2 − ν 2
2
dt
= ν+1
2 F1
3
t
2 Γ(ν + 12 ) 0 eat − 1
2
[
]
πaν
Li2 (e−a ) + aLi1 (e−a ) ,
ν+1
2 Γ(ν + 1)
∑
where the dilogarithm Liα (z) = n=1 z n n−α , |z| 5 1, has the integral representation
∫ ∞ α−1
z
t
Liα (z) =
dt,
ℜ(α) > 0. Γ(α) 0 et − z
+
4. Diﬀerential equations for Kapteyn and Schl¨
omilch series
of modiﬁed Bessel and modiﬁed Struve functions
Kapteyn series of Bessel functions were introduced by Willem Kapteyn
[31,32], and were considered and discussed in details by Nielsen [39] and Watson [61], who devoted a whole section of his celebrated monograph to this
Acta Mathematica Hungarica
16
´ BARICZ and T. K. POGANY
´
A.
theme. Recently, the present authors and Jankov obtained integral representation and ordinary diﬀerential equation descriptions and related results
for real variable Kapteyn series [4,29].
Now, we will consider the Kapteyn series built by modiﬁed Bessel functions of the ﬁrst kind, and modiﬁed Struve functions
Kαν,µ (x) =
∑ αn (
n =1
nµ
)
Iνn (xn) − Lνn (xn) ,
where the parameter space includes positive a > 0, while the sequence
(αn )n=1 ensures the convergence of Kαν,µ (x). Our ﬁrst goal is to establish
a double deﬁnite integral representation formula for Kαν,µ (x). For this goal
we recall the deﬁnition of the conﬂuent Fox–Wright generalized hypergeometric function 1 Ψ∗1 (for the general case p Ψ∗q consult [53, p. 493]):
∗
1 Ψ1
(4.1)
]
[
∞
∑
(a)ρn z n
(a, ρ)
z
=
,
(b, σ)
(b)σn n!
n=0
where a, b ∈ C, ρ, σ > 0 and where, as usual, (λ)µ denotes the Pochhammer
symbol deﬁned, in terms of Euler’s Gamma function, by
{
1,
if µ = 0; λ ∈ C \ {0}
Γ(λ + µ)
=
(λ)µ :=
Γ(λ)
λ(λ + 1) · · · (λ + n − 1), if µ = n ∈ N; λ ∈ C.
The deﬁning series in (4.1) converges in the whole complex z-plane when
∆ = σ − ρ + 1 > 0; if ∆ = 0, then the series converges for |z| < ∇, where
∇ := ρ−ρ σ σ .
Theorem 9. Let µ > ν > 0 and let α ∈ C2 (R+ ), such that α|N = (αn ).
Then for
})
(
{
ν
:= Iα ,
x ∈ 0, 2 min 1,
e lim sup |αn |1/νn
n→∞
we have
Kαν,µ (x)
(4.2)
∫
∞ ∫ [t]
=−
1
0
[(
]
)
α(s)sνs−µ
∂ Γ(νt + 21 ) ( x )νt ⋆ 12 , 12 −
xt
·
d
dt ds.
1 Ψ1
s
(νt, 1)
∂t Γ(νt)
2
Γ(νs + 21 )
Acta Mathematica Hungarica
17
MODIFIED STRUVE FUNCTION
Proof. The Sonin–Gubler formula enables us to transform the summands of the Kapteyn series Kαν,µ (x) into
Kαν,µ+1 (x)
2
=
π
∫
∞
0
∑
n =1
αn
µ−νn
n
(
Jνn (xy)
)
dy.
+ n2 y νn
y2
Making use of the Gegenbauer’s integral expression for Jα [1, p. 204, Eq.
(4.7.5)], after some algebra we get
Kαν,µ+1 (x)
1
=√
π
∫
2
=√
π
1
0
∫
1
√
1 − t2
1
0
{
1
√
1 − t2
2
=√
π
(
)
∑ αn ( x2 1 − t2 )νn ∫
n =1
{
nµ−νn Γ(νn +
1
2
)
∞
0
cos(xty)
dy
y 2 + n2
}
(
)
∑ αn ( x2 1 − t2 )νn e−xtn
n=1
∫
0
1
nµ−νn+1 Γ(νn +
1
2
)
}
dt
dt
1
√
Dα (t) dt,
1 − t2
where the inner sum is evidently the following Dirichlet series
{
(
)}
2 )
(
α
exp
−
n
xt
+
ν
ln
∑ n
x 1−t2
Dα (t) =
,
µ−νn+1
n
Γ(νn + 12 )
n =1
2
and p(t) = xt + ν ln x(1−t
2 ) > 0 for x ∈ (0, 2), since p is increasing on (0, 1).
By the Cauchy convergence test applied to Dα (t) we deduce that
( ex )ν
( ex (
) )ν −xt
1 − t2
e
lim sup |αn |1/n 5
lim sup |αn |1/n < 1,
2ν
2ν
n→∞
n→∞
that is, for all x ∈ Iα the series converges absolutely and uniformly. By the
Cahen formula (1.8) we have
(
)ν ∫ ∫
∞
[z] (( (
) )ν −xt )z
2
x
xt
2
(
)
Dα (t) = ln e
1−t
e
2
x 1 − t2
0
0
· ds
α(s)sνs−µ−1
dz ds.
Γ(νs + 12 )
Acta Mathematica Hungarica
18
´ BARICZ and T. K. POGANY
´
A.
Thus
Kαν,µ+1 (x)
1
= −√
π
∫
∞ ∫ [z]
ds
0
0
α(s)sνs−µ−1
Φν (z) dz ds,
Γ(νs + 12 )
where the t-integral
∫
1
Φν (z) =
0
(
) ν
) )ν −xt )z
ln e−xt ( x2 1 − t2 ) (( x (
√
1 − t2
e
dt
2
1 − t2
has to be evaluated. After indeﬁnite integration, under deﬁnite integral,
expanding the exponential term into Maclaurin series, termwise integration
leads to
∫
( x )νz ∫ 1 (
) νz− 1 −xzt
2
Φν (z) dz =
1 − t2
e
dt
2
0
√
π Γ(νz +
=
2Γ(νz)
√
π Γ(νz +
=
2Γ(νz)
1
2
∞ ( )
2
) ( x )νz ∑
1
2
1
2
) ( x )νz
2
j=0
[
⋆
1 Ψ1
(−xz)j
(νz)j
j!
1
2
j
( 12 , 12 ) − xz
]
(νz, 1)
.
Consequently
]
[
√
1
π ∂ Γ(νz + 2 ) ( x )νz ⋆ ( 12 , 12 ) − xz ,
Φν (z) =
1 Ψ1
(νz, 1)
2 ∂z Γ(νz)
2
and thus
∞ ∫ [t]
∫
Kαν,µ+1 (x)
=−
1
[
× 1 Ψ⋆1
0
∂ Γ(νt + 12 ) ( x )νt
∂t Γ(νt)
2
]
( 21 , 21 ) − xt · d α(s)sνs−µ−1 dt ds.
s
(νt, 1)
Γ(νs + 12 )
Now, our goal is to establish a second order nonhomogeneous ordinary
diﬀerential equation which particular solution is the above introduced special kind Kapteyn series (1.3). Firstly, we introduce the modiﬁed Bessel type
diﬀerential operator
(
)
1 ′
ν2
′′
M [y] ≡ y + y − 1 + 2 y;
x
x
Acta Mathematica Hungarica
19
MODIFIED STRUVE FUNCTION
this operator is associated with the modiﬁed Struve diﬀerential equation,
reads as follows
(
)
ν−1
( x2 )
1 ′
ν2
′′
(4.3)
M [y] ≡ y + y − 1 + 2 y = √
.
x
x
π Γ(ν + 12 )
Theorem 10. Let min (ν, µ) > 0. Then for x ∈ Iα the Kapteyn series
K = Kαν,µ (x) is a particular solution of the nonhomogeneous linear second order ordinary diﬀerential equation
(
)
ν2
1 ′
α
′′
(4.4)
Mµ [K] ≡ K + K − 1 + 2 K
x
x
νn
αn ( x2 )
1 α
2 ∑
= Ξν,µ (x) + √
,
x
x π
Γ(νn + 12 )nµ−νn+1
n =1
where
∫ ∞ ∫ [t]
1
d
∂ Γ(νt + 2 ) ( x )νt
=
dx 1
Γ(νt)
2
1 ∂t
]
[
1 1 α(s)sνs−µ−1 (s − 1)
⋆ ( 2 , 2 )
× 1 Ψ1
dt ds.
− xt · ds
(νt, 1)
Γ(νs + 12 )
Ξαν,µ (x)
Proof. Consider the modiﬁed Struve diﬀerential equation (4.3)
)
(
ν−1
( x2 )
1 ′
ν2
M [y] ≡ y (x) + y (x) − 1 + 2 y(x) = √
x
x
π Γ(ν + 21 )
′′
which possesses the solution y(x) = c1 Iν (x) + c2 Lν (x) + c3 Kν (x), where
Kν stands for the modiﬁed Bessel function of the second kind of order ν. Being Iν and Kν independent particular solutions (the Wronskian
W [Iν , Kν ] = −x−1 ) of the homogeneous modiﬁed Bessel ordinary diﬀerential
equation, which appears on the left side in (4.3), the choice c3 = 0 is legitimate. Thus y(x) = Iνn (x) − Lνn (x) is also a particular solution of (4.3).
Setting ν 7→ νn, we get
(
(
) ′′ 1 (
)′
Iνn (x) − Lνn (x) +
Iνn (x) − Lνn (x)
x
ν 2 n2
− 1+ 2
x
)
(
)
Iνn (x) − Lνn (x) =
νn−1
(x)
.
√ 2
π Γ(νn + 12 )
Acta Mathematica Hungarica
20
´ BARICZ and T. K. POGANY
´
A.
Finally, putting x 7→ xn multiplying the above display with n−µ αn and summing up in n ∈ N, we obtain
νn
[
]
)′
αn ( xn
1( α
2 ∑
2 )
M Kαν,µ = Mµα [K] =
Kν,µ (x) − Kαν,µ+1 (x) + √
,
x
x π
Γ(νn + 12 )nµ+1
n =1
where all three right hand side series converge uniformly inside Iα . Applying the result (4.2) of the previous theorem to the series
Kαν,µ (x) − Kαν,µ+1 (x) =
∑ αn (n − 1) (
n =2
nµ+1
)
Iνn (xn) − Lνn (xn) ,
the summation begins with 2. So, the current lower integration limit in the
Euler–Maclaurin summation formula related to (4.2) becomes 1. By this we
clearify the stated relation (4.4). In the following we concentrate on the summation of Schl¨omilch series (1.6)–(1.7). To unify these procedures, we consider the generalized
e I,L
Schl¨
omilch series (1.5). Obviously TI,L
ν,µ (x), Tν,µ (x) are special cases of
SI,L
ν,µ (x). However, bearing in mind the asymptotics in Sonin–Gubler forI,L
mula (3.1), we see that the necessary
condition
( µ−ν+1
) for the convergence of Sν,µ (x)
for a ﬁxed x > 0 becomes αn = o n
as n → ∞.
Theorem 11. Let min (ν, µ, x) > 0 ∑
and α ∈ C1 (R+ ) be monotone in−µ+ν−1 α converges. Then
creasing such that α|N = (αn )n=1 , and
n
n =1 n
S = SI,L
ν,µ (x) is a particular solution of the nonhomogeneous linear second
order ordinary diﬀerential equation
∑
x ν−1
Mµα [S]
= M[
SI,L
ν,µ
2
(2)
1
′ ν
Υα,2
(x)
−
)
] = x (Υα,1
µ+2 (x) + √
µ+1
2
x
π Γ(ν +
1
2
) n =1
αn
,
µ−ν+1
n
where
∫
Υα,β
µ (x)
∞
=µ
−µt
∫
e
0
[et ]
(
)(
)
du (α(u) uβ − 1 Iν (xu) − Lν (xu) ) dt du.
1
Proof. Consider again the modiﬁed Struve diﬀerential equation (4.3),
which possesses the solution y(x) = c1 Iν (x) + c2 Lν (x) + c3 Kν (x), and choose
the particular solution associated with c1 = −c2 = 1 and c3 = 0. TransformActa Mathematica Hungarica
21
MODIFIED STRUVE FUNCTION
ing (4.3) putting x 7→ xn, multiplying it by αn n−µ and summing up the
equation with respect n ∈ N, we arrive at
)′′
)′ ∑
(∑
(
1 ∑ αn
αn
αn
y(xn) +
y(xn) −
y(xn)
µ
µ+1
n
x
n
nµ
n =1
n=1
n =1
ν−1
( x2 )
ν 2 ∑ αn
− 2
y(xn) = √
x
nµ+2
π Γ(ν +
n =1
Thus
M[
SI,L
ν,µ
1
]= x
(∑
n =2
∑
1
2
) n=1
αn
.
µ−ν+1
n
)′
αn (n − 1)
y(xn)
nµ+1
ν−1
( x2 )
ν 2 ∑ αn (n2 − 1)
− 2
y(xn) + √
x
nµ+2
π Γ( ν +
n =2
∑
1
2
) n =1
αn
.
µ−ν+1
n
Denote
Υα,β
µ (x) =
(4.5)
∑ αn (nβ − 1)
y(xn),
nµ
0 < ν 5 µ, x > 0.
n=2
Following the same lines of the proof of Theorem 7, by Cahen’s formula and
the Euler–Maclaurin summation we immediately yield the double deﬁnite
integral representation
∫ ∞
∫ [et ]
(
)
α,β
−µt
Υµ (x) = µ
e
du (α(u) uβ − 1 y(xu)) dt du;
0
1
which immediatelly lead to the stated result. Corollary 1. Let µ − 1 > ν > 0 and x > 0. Then T = TI,L
ν,µ (x) is a
particular solution of the nonhomogeneous linear second order ordinary differential equation
M [T] =
(4.6)
1 1,1
ζ(µ − ν + 1) ( x )ν−1
ν 2 1,2
′
(x) + √
Υµ+1 (x)) − 2 Υµ+2
,
(
x
x
π Γ(ν + 21 ) 2
where
∫
∞
Υ1,β
µ (x) = µ
0
e−µt
∫
[et ]
(
)(
)
du ( uβ − 1 Iν (xu) − Lν (xu) ) dt du.
1
Acta Mathematica Hungarica
22
´ BARICZ and T. K. POGANY
´
A.
e=T
e I,L
Corollary 2. Let µ > ν > 0 and x > 0. Then T
ν,µ (x) is a particular solution of the nonhomogeneous linear second order ordinary diﬀerential
equation
[ ]
ν2 e 2
η(µ − ν + 1) ( x )ν−1
′
1
e = M [T
e I,L ] = 1 (Υ
e
Mµα T
(x)
Υ
(x)
+
,
−
√
)
ν,µ
x µ+1
x2 µ+2
π Γ(ν + 21 ) 2
where
e β (x) = µ
Υ
µ
∫
∞
−µt
∫
[et ]
e
0
(
)(
)
du (eiπu uβ − 1 Lν (xu) − Iν (xu) ) dt du.
1
Now, a completely diﬀerent type integral representation formula will be
derived for TI,L
ν,ν+1 (x) which simpliﬁes the nonhomogeneous part of related
diﬀerential equation (4.6).
Theorem 12. If ν > 0 and x > 0, then we have
(
)
∫ ∞
1
dt
I,L
Tν,ν+1 (x) =
Jν (xt) coth (πt) −
.
ν+1
πt t
0
Proof. Consider the well-known summation formula [23]
∑
a2
n =1
π
1
1
=
coth (πa) − 2 ,
2
+n
2a
2a
a ̸= in.
In conjunction with the Sonin–Gubler formula (1.8) we conclude that
(∑
)
∫
2 ∞
1
dt
I,L
Jν (xt)
Tν,ν+1 (x) =
2
2
π 0
t + n tν
n =1
∫
(
∞
Jν (xt)
=
0
1
1
coth (πt) − 2
t
πt
)
Remark 1. Actually, the formula
∑
π
1
1
=
coth (πa) − 2 ,
a2 + n2
2a
2a
dt
.
tν
a ̸= 0
n =1
has been considered by Hamburger [23, p. 130, (C)] in the slightly diﬀerent
form
(C)
1+2
∞
∑
−2πna
e
n=1
Acta Mathematica Hungarica
∞
1
2a ∑
1
= i cot πia =
+
,
2
πa
π n=1 a + n2
a ̸= in;
23
MODIFIED STRUVE FUNCTION
Hamburger proved that the functional equation for the Riemann-zeta function is equivalent to (C), see also [11] for connections of the above formulae
to Eisenstein series. Also, it is worth mentioning that further, complex analytical generalizations of the above formula can be found in [8].
5. Novel Bromwich–Wagner line integral representation for Jν (x)
As a by-product of Theorem 12, we arrive at the integral relation
(
)
∑ Iν (xn) − Lν (xn)
I,L
Tν,ν+1 (x) =
,
ν > 0, x > 0
nν+1
n =1
which, in the expanded form reads
(
)
∫ ∞
1
dt
Jν (xt) coth (πt) −
ν+1
πt t
0
∫
∞ ∫ [es ]
= (ν + 1)
0
(
)
e−(ν+1)s du Iν (xu) − Lν (xu) ds du.
0
This arises in a Fredholm type convolutional integral equation of the ﬁrst
kind with degenerate kernel
(
)
∫ ∞
1
dt
(5.1)
f (xt) coth (πt) −
= Fν (x),
ν+1
πt t
0
having nonhomogeneous part
∫ ∞∫
(5.2)
Fν (x) = (ν + 1)
0
[es ]
(
)
e−(ν+1)s du Iν (xu) − Lν (xu) ds du.
0
Obviously, Jν is a particular solution of this equation. Before we state our
result, we say that the functions f and g are orthogonal a.e. ∫with respect to
∞
the ordinary Lebesgue measure on the positive half-line when 0 f (x)g(x) dx
vanishes, writing this as f ⊥ g.
Theorem 13. Let ν > 0 and x > 0. The ﬁrst kind Fredholm type convolutional integral equation with degenerate kernel (5.1) possesses particular
solution f = Jν + h, where h ∈ L1 (R+ ) and
(
)
1
−ν−1
h(x) ⊥ x
coth (πx) −
,
x>0
πx
if and only if the nonhomogeneous part of the integral equation equals Fν (x)
given by (5.2).
Acta Mathematica Hungarica
24
´ BARICZ and T. K. POGANY
´
A.
We mention that h as in the above theorem has been constructed in
[16, Example]. To solve the integral equation (5.2) we use the Mellin integral transform technique, following some lines of a similar procedure used by
Draˇsˇci´c–Pog´
any in [16]. The Mellin transform pair of a suitable f we deﬁne
as [51]
∫ ∞
∫ c+i∞
1
p−1
−1
Mp (f ) :=
x f (x) dx,
Mx (g) :=
x−p Mp (f ) dp,
2πi
0
c−i∞
where the inverse Mellin transform is given in the form of a line integral
with Bromwich–Wagner type integration path which begins at c − i∞ and
terminates at c + i∞. Here the real c belongs to the fundamental strip of
the inverse Mellin transform M −1 .
Theorem 14. Let ν > 0, x > 0. Then the following Bromwich–Wagner
type line integral representation holds true
(5.3)
Jν (x) =
(∫
∫
×
c+i∞
∞ ∫ [es ]
Mp
−(ν+1)s
e
0
c−i∞
du
ν+1
2i
(
)
)
Iν (xu) − Lν (xu) ds du
0
ν−p
Γ( p−ν
2 )Γ( 2 + 1)ζ(ν − p + 2)
xp−1 dp,
where c ∈ (ν, ν + 1).
Proof. Applying M −1 to the equation (5.1), we get
(∫ ∞
)
)
(
1
dt
= Mp (Fν ).
Mp
Jν (xt) coth (πt) −
πt tν+1
0
By the Mellin convolution property
(∫ ∞
)
Mp (f ⋆ g) = Mp
f (rt) · g(t) dt = Mp (f ) · M1−p (g),
0
it follows that
(5.4)
(
)
Mp (x−ν−1 coth πx − (πx)−1 ) · M1−p (Jν ) = Mp (Fν ).
The fundamental analytic strip contains (ν, 1 + ν), because
1 z
 + + O(z 3 ),
if z → 0
coth z = z 3
(
)

1 + 2e−2z [1 + O e−2z ] if z → ∞;
Acta Mathematica Hungarica
25
(
)
in both cases ℜ(z) > 0. Now, rewriting x−ν−1 coth πx − (πx)−1 by (C)
and using termwise the Beta function description, we conclude that
(
)
(
1
p−ν ν−p
−1 )
−ν−1
Mp ( x
coth πx − (πx) ) = B
,
+ 1 ζ(ν − p + 2),
π
2
2
MODIFIED STRUVE FUNCTION
for all p ∈ (ν, ν + 1). Therefore
M1−p (Jν ) =
B(
π Mp (Fν )
,
+ 1)ζ(ν − p + 2)
p−ν ν−p
2 , 2
which ﬁnally results in
Jν (x) =
∫
×
c+i∞
Mp (
∫ ∞ ∫ [es ]
0
0
ν+1
2i
(
)
e−(ν+1)s du Iν (xu) − Lν (xu) ds du)
ν−p
B( p−ν
2 , 2 + 1)ζ(ν − p + 2)
c−i∞
xp−1 dp,
where the fundamental strip contains c = ν + 12 . So, the desired integral
representation formula is established. We note that the formula collection in [24] does not contain (5.3).
Theorem 15. Let 0 < ν < 32 , x > 0. Then
(5.5)
TI,L
ν,ν+1 (x)
=
∫
1
2p+1 πi
Γ( ν−p
2 +
c+i∞
c−i∞
sin [
π
2 (p
)ζ(ν − p + 2) −p
x dp.
1
− ν)] · Γ( ν+p
2 + 2)
1
2
Here also c ∈ (ν, ν + 1).
Proof. Consider relation (5.4). Expressing M1−p (Jν ) via formula [41,
p. 93, Eq. 10.1]
ν+p
(
)
2p−1 Γ( 2 )
Mp Jν (ax) = p
,
a Γ( ν−p
2 + 1)
3
a > 0, −ν < p < ,
2
equality (5.4), by virtue of the Euler’s reﬂection formula becomes
Mp (Fν ) =
ν−p
Γ( ν−p
2 + 1)Γ( 2 +
2p πΓ
=
Γ( ν−p
2 +
2p sin π2 (p
[
(
p−ν
1
2 Γ
2
ν+p
1
+
2
2
) (
)ζ(ν − p + 2)
)
)ζ(ν − p + 2)
.
1
− ν)] · Γ( ν+p
2 + 2)
1
2
Acta Mathematica Hungarica
26
´ BARICZ and T. K. POGANY
´
A.
Having in mind that Fν (x) is the integral representation of the Schl¨omilch
−1 we arrive at the asserted
series TI,L
ν,ν+1 (x), inverting the last display by Mp
result. Acknowledgements. The authors cordially thank anonymous referee
´ Baricz was
for constructive comments and suggestions. The research of A.
supported by the Romanian National Council, project number PN-II-RUTE 190/2013.
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