4_792_pogany.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 980-988
ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Mathieu–type Series for the ℵ–function Occurring in

Fokker–Planck Equation

Ram K. Saxena 1, Tibor K. Pogány 2,∗

1 Department of Mathematics and Statistics, Jain Narain Vyas University, Jodhpur–342004, India
2 Faculty of Maritime Studies, University of Rijeka, Studentska 2, 51000 Rijeka, Croatia

Abstract. Closed form expressions are obtained for a family of convergent Mathieu type a–series and
its alternating variant, whose terms contain an ℵ–function, which naturally occurs in certain problems
associated with driftless Fokker–Planck equation with power law diffusion [25]. The ℵ–function is a
generalization of the familiar H–function and the I–function. The results derived are of general char-
acter and provide an elegant generalization for the closed form expressions the Mathieu–type series
associated with the H–function by Pogány [8], for Fox–Wright functions by Pogány and Srivastava [13]
and for generalized hypergeometric pFq and Meijer’s G–function by Pogány and Tomovski [16], and

others. For the H–function [7, p. 216] the results are obtained very recently by Pogány and Saxena
[11].

2000 Mathematics Subject Classifications: Primary 33C20, 33C60; Secondary 40G99, 44A20.

Key Words and Phrases: I–function, Dirichlet series, H–function, Fox–Wright function, Laplace inte-
gral representation for Dirichlet series, Mathieu a–series, Mellin–Barnes type integrals, Mittag–Leffler
function

1. Introduction and Preliminaries

In order to unify and extend the results for the convergent Mathieu–type a–series and
its alternating variants whose terms contain the familiar transcendental functions, such as
Gauss hypergeometirc function 2F1, generalized hypergeometric function pFq , the Fox–Wright

∗Corresponding author.

Email addresses: poganj�pfri.hr (T. Pogány), ram.saxena�yahoo.
om (R. Saxena)

http://www.ejpam.com 980 c© 2010 EJPAM All rights reserved.



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 981

function, the Meijer’s G–function and Fox’s H–function published in a series of papers by
Pogány [8, 9, 10], Pogány et al. [11, 12, 13, 14, 15, 16] and Srivastava and Tomovski [23].

Inequalities and integral representations for Mathieu–type series are discussed by Cerone
and Lenard [1], Pogány and Tomovski [17], Srivastava and Tomovski [23] and others. The
results obtained by the authors in this serve as the key formulæ for numerous potentially
useful special functions of Science, Engineering and Technology scattered in the literature.

In the study of fractional driftless Fokker–Planck equations with power law diffusion co-
efficients, there arises naturally a special function, which is a special case of the ℵ, that is
Aleph–function. The idea to introduce Aleph–function belongs to Südland et al. [24], how-
ever the notation and complete definition is presented here in the following manner in terms
of the Mellin–Barnes type integrals [also see 25]:

ℵ[z] = ℵ m,n
pi ,qi,τi ;r

[z] = ℵ m,n
pi ,qi ,τi;r

�
z

�����
(a j,A j)1,n, . . . , [τ j(a j ,A j)]n+1,pi

(b j, B j)1,m, . . . , [τ j(b j, B j)]m+1,qi

�

:=
1

2πω

∫

L

Ω m,n
pi ,qi ,τi;r

(s)z−s ds (1)

for all z 6= 0, where ω=
p−1 and

Ω m,n
pi ,qi ,τi ;r

(s) =

m∏
j=1
Γ(b j + B js) ·

n∏
j=1
Γ(1− a j − A js)

r∑
i=1
τi

pi∏
j=n+1

Γ(a ji + A jis) ·
qi∏

j=m+1
Γ(1− b ji − B jis)

, (2)

The integration path L = Liγ∞,γ ∈ R extends from γ− i∞ to γ+ i∞, and is such that the

poles, assumed to be simple, of Γ(1− a j − A js), j = 1, n do not coincide with the poles of

Γ(b j + B js), j = 1, m. The parameters pi ,qi are non–negative integers satisfying 0 ≤ n ≤ pi ,

1 ≤ m ≤ qi,τi > 0 for i = 1, r. The parameters A j, B j ,A ji , B ji > 0 and a j, b j ,A ji , b ji ∈ C. The
empty product in (2) is interpreted as unity. The existence conditions for the defining integral
(1) are given below:

ϕℓ > 0, |arg(z)| < π
2
ϕℓ ℓ = 1, r; (3)

ϕℓ ≥ 0, |arg(z)| < π
2
ϕℓ and ℜ{ζℓ}+ 1< 0 , (4)

where

ϕℓ =

n∑

j=1

A j +

m∑

j=1

B j −τℓ
� pℓ∑

j=n+1

A jℓ+

qℓ∑

j=m+1

B jℓ

�
(5)

ζℓ =

m∑

j=1

b j −
n∑

j=1

a j +τℓ

� qℓ∑

j=m+1

b jℓ−
pℓ∑

j=n+1

a jℓ

�
+

1

2

�
pℓ− qℓ
�

ℓ= 1, r. (6)



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 982

Remark 1. If the sum in the denominator of (2) can be simplified in terms of a polynomial in s,

the factors of this polynomial can be expressed by a fraction of Euler’s Gamma function leading

to an H–function instead, see [25, p. 325].

Remark 2. It is observed that there is no historical name given to (1), compared to [24]. The

Mellin transform of this function is the coefficient of z−s in the integrand of (1). There are no

references containing tables of ℵ–functions in the literature.

For τ1 = τ2 = . . . = τr = 1, in (1) the definition of following I–function [21] is recovered:

I[z] = ℵ m,n
pi ,qi ,1;r[z] = ℵ m,n

pi ,qi ,1;r

�
z

�����
(a j ,A j)1,n, . . . , (a j ,A j)n+1,pi

(b j, B j)1,m, . . . , (b j, B j)m+1,qi

�

:=
1

2πω

∫

L

Ω
m,n

pi ,qi ,1;r(s)z
−s ds, (7)

where Ω m,n
pi ,qi ,1;r(s) is defined in (2). The existence conditions for the integral in (7) are the

same as given in (3)–(6) with τi = 1, i = 1, r.
If we further set r = 1, then (7) reduces to the familiar H–function given e.g. in the

monograph [7]:

Hm,n
p,q [z] = ℵ m,n

pi ,qi ,1;1[z] = ℵ m,n
pi ,qi ,τi;1

�
z

�����
(ap,Ap)

(bq, Bq)

�
:=

1

2πω

∫

L

Ω
m,n

pi ,qi ,1;1(s)z
−s ds , (8)

where the kernel Ω m,n
pi ,qi ,1;1(s) is given in (2), which itself is a generalization of Meijer’s G–

function [2, p. 207] to which it reduces for A1 = . . . = Ap = 1 = B1 = . . . = Bq. A detailed
and comprehensive account of the H–function is available from the monographs written by
Mathai and Saxena [6], Srivastava et al. [22], Kilbas and Saigo [4] and Mathai et al. [7].

In what follows, the Aleph function will be represented by the contracted notations
ℵ m,n

pi ,qi,τi ;r
[z] or ℵ[z].

Now, consider the Mathieu–type a–series Θλ,µ and its alternating variant eΘλ,µ, defined by

Θλ,µ

n
ℵ;c, x
o

:=
∞∑

j=1

ℵ m,n+1
pi+1,qi ,τi ;r

h x
c j

��� (α,β), (a j,A j)1,n, (a ji,A ji)n+1,pi

(b j, B j)1,m, (b ji, B ji)m+1,qi

i

cλ
j
(c j + x)µ

, (9)

eΘλ,µ

n
ℵ;c, x
o

:=
∞∑

j=1

(−1) j−1ℵ m,n+1
pi+1,qi ,τi ;r

h x
c j

��� (α,β), (a j,A j)1,n, (a ji,A ji)n+1,pi

(b j, B j)1,m, (b ji, B ji)m+1,qi

i

cλ
j
(c j + x)µ

(10)

where the convention is followed that the positive sequence c =
�
cn

�
n∈N monotonously in-

creases and tends to infinity; equivalently

c : 0< c1 < c2 < . . . < cn ↑ ∞ . (11)



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 983

2. Integral Representations of Θλ,µ

n
ℵ;c, x
o

and eΘλ,µ

n
ℵ;c, x
o

The Laplace transform of the ℵ–function can be established in the following form

∫ ∞

0

xλ−1e−sxℵ m,n
pi ,qi ,τi ;r

�
ηxρ
�

dx

= s−λℵ m,n+1
pi+1,qi ,τi ;r

h η
sρ

��� (1−λ,ρ), (a j,A j)1,n, [τi(a ji,A ji)]n+1,pi

(b j, B j)1,m, [τi(b ji, B ji)]m+1,qi

i
, (12)

where λ, s,η ∈ C; ℜ{s} > 0, ρ > 0,τi > 0, i = 1, r, and

ℜ{λ}+ρ min
1≤ j≤m

ℜ{b j}
B j

> 0, |arg(η)| < π
2

min
1≤ℓ≤r

�
ζℓ
�

; (13)

the parameter ζℓ is defined in (6).
The formula (12) can be easily established with the help of the definition (2) of ℵ–function

and using gamma function formula

Γ(µ)ζ−µ =
∫ ∞

0

xµ−1 e−ζxdx min
�
ℜ{µ},ℜ{ζ}
�
> 0 .

Theorem 1. Let λ > 0,µ > 0, x > 0,α = 1− λ,β = ρ and let the sequence c satisfies (11).
Then there hold the following results:

Θλ,µ

n
ℵ;c, x
o
= I
ℵ
c (λ+ 1,µ) +µIℵc (λ,µ+ 1) (14)

eΘλ,µ

n
ℵ;c, x
o
= eIℵc (λ+ 1,µ) +µeIℵc (λ,µ+ 1) , (15)

where

I
ℵ
c (u, v) :=

∫ ∞

c1

�
c−1(t)
�

tu(t + x)v
ℵ m,n+1

pi+1,qi ,τi;r

h x
t

; u
i

dt , (16)

eIℵc (u, v) :=

∫ ∞

c1

sin2 �π
2
[c−1(t)]
�

tu(t + x)v
ℵ m,n+1

pi+1,qi ,τi ;r

h x
t

; u
i

dt, (17)

and

ℵ m,n+1
pi+1,qi ,τi ;r

h x
t

; u
i

:= ℵ m,n+1
pi+1,qi ,τi ;r

h x
t

��� (1− u, 1), (a j ,A j)1,n, [τi(a ji,A ji)]n+1,pi

(b j, B j)1,m, [τi(b ji, B ji)]m+1,qi

i
.

where c : R+ 7→ R+ is an increasing function such that c(x)
��
x∈N = c, c−1(x) is the inverse of

c(x),
�

c−1(x)
�

stands for the integer part of the quantity c−1(x).



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 984

Proof. Taking ζ= cn+ x in (14), setting s = c j; ρ = 1,η = x and inserting α= 1−λ,β = 1
in (12), we find that

Θλ,µ

n
ℵ;c, x
o
=

∞∑

j=1

ℵ m,n+1
pi+1,qi ,τi ;r

h x
c j

��� (1−λ, 1), (a j,A j)1,n, [τi(a ji,A ji)]n+1,pi

(b j, B j)1,m, [τi(b ji, B ji)]m+1,qi

i

cλ
j
(c j + x)µ

=
1

Γ(µ)

∞∑

j=1

∫ ∞

0

sλ−1e−c jsℵ m,n
pi ,qi ,τi;r

[xs]ds

∫ ∞

0

tµ−1 e−(c j+x)t dt

=
1

Γ(µ)

∫ ∞

0

∫ ∞

0

� ∞∑

j=1

e−c j(s+t)

�
sλ−1 tµ−1e−x tℵ m,n

pi ,qi ,τi ;r
[xs]ds dt , (18)

where, by convergence reasons µ > 0 is already assumed. Following the lines of the use of
Dirichlet series technique used in earlier papers by Pogány and coworkers [10, 11, 12, 13, 14,
15, 16], by means of (12) we conclude

Θλ,µ

n
ℵ;c, x
o
=

1

Γ(µ)

∫ ∞

0

∫ ∞

0

∫ ∞

c1

sλ tµ−1e−(y+x)t−ysℵ m,n
pi ,qi,τi ;r

[xs]
�

c−1(y)
�

dsdtdy

+
1

Γ(µ)

∫ ∞

0

∫ ∞

0

∫ ∞

c1

sλ−1 tµe−(y+x)t−ysℵ m,n
pi ,qi ,τi ;r

[xs]
�

c−1(y)
�

dsdtdy := Js + Jt .

Introducing the auxiliary integral

I
ℵ
c (u, v) :=

∫ ∞

c1

�
c−1(t)
�

tu(t + x)v
ℵ m,n+1

pi+1,qi ,τi ;r

h x
t

; u
i

dt ,

it readily follows that

Js = I
ℵ
c (λ+ 1,µ) and Jt = µ · Iℵc (λ,µ+ 1) .

This finishes the proof of (16).
The proof of (17) is similar to that of (16), if we employ the definition of the new alter-

nating inner Dirichlet series eDc(·) [14, p. 77, Section 4] given below:

eDc(s+ t) =

∞∑

j=1

(−1) j−1e−c j(s+t) = (s+ t)

∫ ∞

c1

e−(s+t)x sin2
�π

2

�
c−1(x)
��

dx . (19)

The application of (19) completes the proof of (17).

3. Special Cases

As Aleph function is the most generalized special function, numerous special cases with
potentially useful transcendental functions, such Mittag–Leffler functions, Bessel functions,



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 985

Whittaker functions, hypergeometric functions, generalized hypergeoemtric pFq function, Mei-
jer’s G–function, Fox–Wright Ψ function and Fox H–function and their special cases can be
deduced by making suitable changes in the parameters. But, for the sake of brevity, some
interesting special cases of Theorem 1 are given below.

Corollary 1. [12, Theorem] Let λ,µ, x > 0,α = 1− λ,β = ρ,τ1 = . . . = τr = 1, and let the

sequence c satisfies (11). Then the Aleph function reduces to an I–function and there holds the

following result

Θλ,µ

n
I

m,n+1
pi+1,qi ,r

;c, x
o
= I

I
c(λ+ 1,µ)+µII

c(λ,µ+ 1) (20)

eΘλ,µ

n
I

m,n+1
pi+1,qi ,r

;c, x
o
= eII

c(λ+ 1,µ)+µeII
c(λ,µ+ 1) , (21)

where

I
I
c(u, v) :=

∫ ∞

c1

�
c−1(t)
�

tu(t + x)v
I

m,n+1
pi+1,qi ,r

h x
t

; u
i

dt , (22)

eII
c(u, v) :=

∫ ∞

c1

sin2 �π
2
[c−1(t)]
�

tu(t + x)v
I

m,n+1
pi+1,qi ,r

h x
t

; u
i

dt, (23)

with

I
m,n+1

pi+1,qi ,r

h x
t

; u
i

:= ℵ m,n+1
pi+1,qi ,1;r

h x
t

��� (1− u, 1), (a j,A j)1,n, (a ji,A ji)n+1,pi

(b j, B j)1,m, (b ji, B ji)m+1,qi

i
.

Here c remains the same as above in Theorem 1.

When r = 1,τ1 = 1, the Aleph function reduces to Fox’s H–function and Theorem 1 gives
rise to the following result given by Pogány [10, Theorem].

Corollary 2. Let λ,µ, r > 0,α = 1− λ,β = ρ = 1 and let the sequence c satisfies the condition

given in (11). Then we have

Θλ,µ

n
H

m,n+1
p+1,q ;c, x
o
= I

H
c (λ+ 1,µ) +µIH

c (λ,µ+ 1) (24)

eΘλ,µ

n
H

m,n+1
p+1,q ;c, x
o
= eIH

c (λ+ 1,µ) +µeIH
c (λ,µ+ 1) , (25)

where

I
H
c (u, v) :=

∫ ∞

c1

�
c−1(t)
�

tu(t + x)v
H

m,n+1
p+1,q

h x
t

���
(1− u, 1), (a j,A j)1,n, (a j,A j)n+1,p

(b j, B j)1,m, (b j, B j)m+1,q

i
dt (26)

eIH
c (u, v) :=

∫ ∞

c1

sin2 �π
2
[c−1(t) ]
�

tu(t + x)v
H

m,n+1
p+1,q

h x
t

���
(1− u, 1), (a j ,A j)1,n, (a j,A j)n+1,p

(b j, B j)1,m, (b j, B j)m+1,q

i
dt. (27)

The Fox–Wright function pΨq is defined [7, p. 23] by the power series in the form

pΨq

h (ap,αp)

(bq,βq)

��� z
i

:=
∞∑

n=0

∏p

j=1 Γ
�
a j + A jn
�

∏q

j=1 Γ
�

b j + B jn
�

zn

n!
. (28)



R. Saxena, T. Pogány / Eur. J. Pure Appl. Math, 3 (2010), 980-988 986

with a j , b j ∈ C, A j, B j ∈ R,Ai · B j 6= 0 (i = 1, p, j = 1,q) and
∑q

j=1 B j −
∑p

j=1 A j > −1. Now,
employing the identity [7, p. 25]

pΨq

h (ap,Ap)

(bq, Bq)

��� − z
i
= H

1,p
p,q+1

h
z

��� (1− ap,Ap)

(0,1), (1− bq, Bq)

i
, (29)

pointing out that we can express it via the Aleph function as

pΨq

h (ap,Ap)

(bq, Bq)

��� − z
i
= ℵ 1,p

p,q+1,1;1

h
z

��� (1− ap,Ap)

(0,1), (1− bq, Bq)

i
,

it is not difficult to deduce the corresponding results for Fox–Wright function.
Let us define

Θλ,µ
�

p+1Ψq;c, x
	

:=
∞∑

j=1

p+1Ψq

h (α,β), (ap,Ap)

(bq, Bq)

��� − x

c j

i

cλ
j
(c j + x)µ

, (30)

and

eΘλ,µ
�

p+1Ψq;c, x
	

:=
∞∑

j=1

(−1) j−1
p+1Ψq

h (α,β), (ap,Ap)

(bq, Bq)

��� − x

c j

i

cλ
j
(c j + x)µ

. (31)

We then obtain the following

Corollary 3. Let λ 6∈ N,µ > 0, r > 0, (α,β) = (1−λ, 1), (bq,βq) = (1,1) and let the sequence c

satisfies (11). Then we have

Θλ,µ
�

p+1Ψq;c, r
	
= I

Ψ
c (λ+ 1,µ) +µIΨc (λ,µ+ 1) (32)

eΘλ,µ
�

p+1Ψq;c, r
	
= eIΨc (λ+ 1,µ) +µeIΨc (λ,µ+ 1) , (33)

where

I
Ψ
c (u, v) :=

∫ ∞

c1

�
c−1(t)
�

tu(t + x)v
· p+1Ψq

h (1− u, 1), (ap,Ap)

(1,1), (bq−1, Bq−1)

��� − x

t

i
dt (34)

and

eIΨc (u, v) :=

∫ ∞

c1

sin2 �π
2
[c−1(t) ]
�

tu(t + x)v
· p+1Ψq

h (1− u, 1), (ap,Ap)

(1,1), (bq−1, Bq−1)

��� − x

t

i
dt . (35)

Remark 3. Finally, it is interesting to observe that by virtue of the relation

Eα,β(z) = H
1,1
1,2

h
− z

��� (0,1)
(0,1), (1− β ,α)

i
= ℵ 1,1

1,2,1;1

h
− z

��� (0,1)
(0,1), (1− β ,α)

i
,

where Eα,β(z) is the Mittag–Leffler function [Chapter 18 3] and [p. 80 5], defined by

Eα,β(z) =

∞∑

n=1

zn

Γ(αn+ β)
,

where α,β ∈ C;ℜ{α},ℜ{β} > 0, the similar type of results for the Mittag–Leffler function can

be deduced from Corollary 2.



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