Certain interesting single (or double) infinite series associated with hypergeometric functions have been expressed in terms of Psi (or Digamma) function 𝜓(
z
), for example, see Nishimoto and Srivastava
[8]
, Srivastava and Nishimoto
[13]
, Saxena
[10]
, and Chen and Srivastava
[5]
, and so on. In this sequel, with a view to unifying and extending those earlier results, we
first
establish two relations which some double infinite series involving hypergeometric functions are expressed in a single infinite series involving
𝜓
(
z
). With the help of those series relations we derived, we
next
present two functional relations which some double infinite series involving
-functions, which are defined by a generalized Mellin-Barnes type of contour in- tegral, are expressed in a single infinite series involving 𝜓(
z
). The results obtained here are of general character and only two of their special cases, among numerous ones, are pointed out to reduce to some known results.
1. INTRODUCTION AND PRELIMINARIES
Certain interesting single (or double) infinite series associated with hypergeometric functions (1.4) have recently been expressed in terms of Psi (or Digamma) function 𝜓(
z
) in (1.1), for example, see Nishimoto and Srivastava
[8]
, Srivastava and Nishimoto
[13]
, Saxena
[10]
, Chen and Srivastava
[5]
and Srivastava and Choi
[15]
, and so on. In this connection, with a view to unifying and extending those earlier results, we
first
establish two relations which some double infinite series involving hypergeometric functions are expressed in a single infinite series involving 𝜓(
z
). With the help of those series relations we derived, we
next
present two functional relations which some double infinite series involving H-functions in (3.1), which are defined by a generalized Mellin-Barnes type of contour integral, are expressed in a single infinite series involving 𝜓(
z
). The results obtained here are of general character and only two of their special cases, among numerous ones, are pointed out to reduce to some known results.
To do this, we begin by recalling the Psi (or Digamma) function 𝜓(
z
) (cf.
[15, Section 1.2]
and
[16, p. 24]
) defined by
and the following well-known (rather classical) result (see, for example,
[16, p. 352]
):
where Γ is the familiar Gamma function, (λ)
n
denotes the Pochhammer symbol defined (for λ ∈ ℂ) by
and ℂ and
are the sets of complex numbers and nonpositive integers, respectively.
A natural generalization of the hypergeometric functions
2
F
1
,
1
F
1
,
et cetera
(considered in the vast literature; see, for example,
[16, p. 71]
) is accomplished by the introduction of an arbitrary number of numerator and denominator parameters. The resulting series:
where (λ)
n
is the Pochhammer symbol defined by (1.3), is known as the
generalized Gauss
(
and Kummer
)
series
, or simply, the
generalized hypergeometric series
.
The summation formula (1.2) and its obvious special cases were revived, in recent years, as illustrations emphasizing the usefulness of fractional calculus in evaluating infinite sums. For a detailed historical account of (1.2), and of its various consequences and generalizations have been presented by Nishimoto and Srivastava
[8]
. A systematic account of certain family of infinite series which can be expressed in terms of Digamma functions together with their relevant unification and generalization has been given by Srivastava
[14]
, Al-Saqabi
et al
.
[1]
and Aular de Duran
et al
.
[2]
.
From the aforementioned work of Nishimoto and Srivastava
[8]
, we choose to recall here two interesting consequences of the summation formula (1.2), which are contained in Theorem 1 below.
Theorem 1
(
[8]
).
Let
be an arbitrary bounded sequence of complex numbers. Then we have
and
provided that each of the series involved converges absolutely.
2. GENERALIZATIONS OF THE RESULTS IN THEOREM 1
In this section, we establish certain generalizations of the formulas (1.5) and (1.6).
Theorem 2.
Let
be an arbitrary bounded sequence of complex numbers and set
Then we obtain
provided that each of the series involved converges absolutely.
Theorem 3.
Let
be an arbitrary bounded sequence of complex numbers and set
Then we get
provided that each of the series involved converges absolutely.
Proof of Theorems 2 and 3
. For sake of convenience, let the left-hand side of the (2.2) be denoted by
. Then, substituting for
Un
from (2.1) and applying the definitions (1.3) and (1.4), we have
where the inversion of the order of summation can be justified by the absolute convergence of the series involved. The innermost series in (2.4) is summable by means of the well-known result (1.2). We thus have
provided that ℜ(
μ
+
β
) > 0,
, ℜ (
ρ
(
k
)) ≥ 0 for all
Now we have
Upon using the following known summation formula
[5, p. 380, Eq. (2.5)]
:
and Gauss’s well-known summation theorem for
2
F
1
(
a; b
;
c
; 1) (see,
e.g
.,
[16, p. 64, Eq. (7)]
; see also
[12]
), after a little simplification, we are easily led to the desired result (2.2).□
The equality (2.3) in Theorem 3 will be established in a similar way as in the proof of equality (2.2).
Remark.
The results [14, Theorem 3] look very similar to those in Theorems 2 and 3 here. Yet, it is easy to see that the results in Theorems 2 and 3 here are neither special nor general cases of those in [14, Theorem 3] land vice versa.
3. DEFINITION AND EXISTENCE CONDITIONS OF-FUNCTION
A lot of research work has recently come up on the study and development of a function that is more general than the Fox
H
-function (see,
e.g
.,
[10
,
11]
), popularly known as
-function. It was introduced by Inayat-Hussain
[6
,
7]
and now stands on a fairly firm footing through the following contributions of various authors
[3
,
4
,
6
,
7
,
9
,
10]
.
The
-function is defined and represented in the following manner
[6]
:
where
z
≠ 0 and
It may be noted that the
contains fractional powers of some of the Gamma functions. Here
z
may be real or complex but is not equal to zero, and an empty product is interpreted as unity;
m
,
n
,
p
, and
q
are integers such that 1 ≤
m
≤
q
, 0 ≤
n
≤
p
; α
j
> 0 (
j
= 1,...,
p
),
βj
> 0 (
j
= 1,...,
q
) and
aj
(
j
= 1,...,
p
) and
bj
(
j
= 1,...,
q
) are complex numbers. The exponents
Aj
(
j
= 1,...,
n
) and
Bj
(
j
=
m
+ 1;...,
q
) take on non-integer values.
The nature of the contour
L
, sufficient conditions of convergence of defining integral (3.1) and other details about the
-function can be seen in
[4
,
6
,
7]
.
The behavior of the
-function for small values of │
z
│ follows easily from a result given by Rathie
[9]
:
where
The following series representation for the
-function given by Saxena
et al
.
[11]
will be required later on:
where
and
The function
makes sense and defines an analytic function of
z
in the following two cases
[3]
:
(i) 0 < │z│ < ∞ and
(ii) μ1 = 0, 0 < │z│ τ˗1 and
4. FUNCTIONAL RELATIONS INVOLVING GENERALIZED MELLIN-BARNES TYPE OF CONTOUR INTEGRAL
Here we give two interesting double summation formulas involving the
-function asserted by the following theorem.
Theorem 4.
If each of the series involved converges absolutely, the following formulas hold
:
and
where
C
> 0, ℜ(
μ
+
β
) > 0, ℜ(
μ
+
η
) > 0,
and
and
ς
(
h
;
k
)
are given in
(3.4) and (3.5),
respectively.
Proof
. In view of the
-function representation (3.3), we apply Theorem 2 by setting
ρ
≡
ρ
(
k
) =
C
ς
(
h, k
) (
C
> 0) and
, where
and
ς
(
h, k
) are defined by (3.4) and (3.5), respectively. Then we have
Now, replacing
z
by
z
1=Bh
in (4.3) and multiplying each side of equality (4.3) by
zbh
=Bh
, then summing both sides of the resulting equations from
h
= 1 to
h
=
m
(≤
q
), we get
This, in view of (3.3), proves the required result (4.1).
A similar argument as in the proof of (4.1) will establish the formula (4.2). This completes the proof of Theorem 4.
5. SPECIAL CASES AND CONCLUDING REMARKS
In this section we briefly consider another variation of the results derived in the preceding sections. On account of the most general nature of the
-function in our main results given by (4.1) and (4.2), a large number of infinite series relations involving simpler functions can be easily obtained as their special cases. Yet, as an illustration, a few interesting special cases will be considered as follows:
-
(i) Forα+β=ν,η= ˗β, andμ= λ ˗ν, the-function reduces to the familiar FoxH-function. Then the functional relations (4.1) and (4.2) yield equalities (3.10) and (3.11) in Chen and Srivastava[5, p. 385].
-
(ii) If we setη= 1 in (4.1) and (4.2) and give some suitable parametric replacement in the resulting identities, we can arrive at the equalities (4.1) and (4.2) in Saxena[10, pp. 128-129].
-
(iii) If we setα=ν,μ= λ ˗α,η= ˗βandν+β=αin Theorems 2 and 3, we are led to Theorems 3 and 4 in[5], respectively.
Theorem 4 gives a further generalization of the functional relations (3.10) and (3.11) in
[5, p. 385]
.
Acknowledgements
The authors should like to express their deep gratitude for the reviewers's very helpful comments so that this paper can be further improved in the present form.
Al-Saqabi B.N
,
Kalla S.L
,
Srivastava H. M
1991
A certain family of infinite series associated with Digamma functions
J. Math. Anal. Appl.
159
361 -
372
DOI : 10.1016/0022-247X(91)90200-J
Aular de Duran J
,
Kalla S.L
,
Srivastava H.M
1995
Fractional calculus and the sums of certain families of infinite series
J. Math. Anal. Appl.
190
738 -
754
DOI : 10.1006/jmaa.1995.1107
Braaksma B.L.J
1964
Asymptotic expansions and analytic continuations for a class of Barnes-integrals
Compositio Math.
15
239 -
341
Buschman R.G
,
Srivastava H.M
1990
The H-function associated with a certain class of Feyman integrals
J. Phys. A: Math. Gen.
23
4707 -
4710
DOI : 10.1088/0305-4470/23/20/030
Chen K.Y
,
Srivastava H.M
2000
Some infinite series and functional relations that arose in the context of fractional calculus
J. Math. Anal. Appl.
252
376 -
388
DOI : 10.1006/jmaa.2000.7079
Inayat-Hussain A.A
1987
New properties of hypergeometric series derivable from Feynman integrals: I. Transformation and reduction formulae
J. Phys. A. : Math. Gen.
20
4109 -
4117
DOI : 10.1088/0305-4470/20/13/019
1987
New properties of hypergeometric series derivable from Feynman integrals: II. A generalisation of the H function
J. Phys. A.: Math. Gen.
20
4119 -
4128
DOI : 10.1088/0305-4470/20/13/020
Nishimoto K
,
Srivastava H.M
1989
Certain classes of infinite series summable by means of fractional calculus
J. College Engrg. Nihon Univ. Ser. B
30
97 -
106
Rathie A.K
1997
A new generalization of generalized hypergeometric functions
Le mathematiche Fasc. II
52
297 -
310
Saxena R.K
1998
Functional relations involving generalized H-function
Le Mathematiche Fasc. I
53
123 -
131
Saxena R.K
,
Ram C
,
Kalla S.L
2002
Applications of generalized H-function in bivariate distributions
Rew. Acad. Canar. Cienc. XIV (Nu’ms.1-2)
(1-2)
111 -
120
Srivastava H.M
1992
A simple algorithm for the evaluation of a class of generalized hyper-geometric series
Stud. Appl. Math.
86
79 -
86
Srivastava H.M
,
Nishimoto K
1992
An elementary proof of a generalization of certain functional relation
J. Fractional Calculus
1
69 -
74
Srivastava R
1991
A simplified overview of certain relations among infinite series that arose in the contex of fractional calculus
J. Math. Anal. Appl.
162
152 -
158
DOI : 10.1016/0022-247X(91)90183-Z
Srivastava H.M
,
Choi J
2001
Series Associated with the Zeta and Related Functions
Kluwer Academic Publishers
Dordrecht, Boston, and London
2012
Zeta and q-Zeta Functions and Associated Series and Integrals
Elsevier Science Publishers
Amsterdam;London;New York