Advanced
CERTAIN CLASSES OF INFINITE SERIES DEDUCIBLE FROM MELLIN-BARNES TYPE OF CONTOUR INTEGRALS
CERTAIN CLASSES OF INFINITE SERIES DEDUCIBLE FROM MELLIN-BARNES TYPE OF CONTOUR INTEGRALS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2013. Oct, 20(4): 233-242
Copyright © 2013, Korean Society of Mathematical Education
  • Received : March 08, 2013
  • Accepted : October 01, 2013
  • Published : October 30, 2013
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
JUNESANG CHOI
DEPARTMENT OF MATHEMATICS, DONGGUK UNIVERSITY, GYEONGJU 780-714, REPUBLIC OF KOREAEmail address:junesang@mail.dongguk.ac.kr
PRAVEEN AGARWAL
DEPARTMENT OF MATHEMATICS, ANAND INTERNATIONAL COLLEGE OF ENGINEERING, JAIPUR303012, RAJASTHAN, INDIAEmail address:goyal.praveen2011@gmail.com

Abstract
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.
Keywords
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
PPT Slide
Lager Image
and the following well-known (rather classical) result (see, for example, [16, p. 352] ):
PPT Slide
Lager Image
where Γ is the familiar Gamma function, (λ) n denotes the Pochhammer symbol defined (for λ ∈ ℂ) by
PPT Slide
Lager Image
and ℂ and
PPT Slide
Lager Image
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:
PPT Slide
Lager Image
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
PPT Slide
Lager Image
be an arbitrary bounded sequence of complex numbers. Then we have
PPT Slide
Lager Image
and
PPT Slide
Lager Image
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
PPT Slide
Lager Image
be an arbitrary bounded sequence of complex numbers and set
PPT Slide
Lager Image
Then we obtain
PPT Slide
Lager Image
provided that each of the series involved converges absolutely.
Theorem 3. Let
PPT Slide
Lager Image
be an arbitrary bounded sequence of complex numbers and set
PPT Slide
Lager Image
Then we get
PPT Slide
Lager Image
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
PPT Slide
Lager Image
. Then, substituting for Un from (2.1) and applying the definitions (1.3) and (1.4), we have
PPT Slide
Lager Image
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
PPT Slide
Lager Image
provided that ℜ( μ + β ) > 0,
PPT Slide
Lager Image
, ℜ ( ρ ( k )) ≥ 0 for all
PPT Slide
Lager Image
Now we have
PPT Slide
Lager Image
Upon using the following known summation formula [5, p. 380, Eq. (2.5)] :
PPT Slide
Lager Image
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
PPT Slide
Lager Image
-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
PPT Slide
Lager Image
-function is defined and represented in the following manner [6] :
PPT Slide
Lager Image
where z ≠ 0 and
PPT Slide
Lager Image
It may be noted that the
PPT Slide
Lager Image
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
PPT Slide
Lager Image
-function can be seen in [4 , 6 , 7] .
The behavior of the
PPT Slide
Lager Image
-function for small values of │ z │ follows easily from a result given by Rathie [9] :
PPT Slide
Lager Image
where
PPT Slide
Lager Image
The following series representation for the
PPT Slide
Lager Image
-function given by Saxena et al . [11] will be required later on:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
The function
PPT Slide
Lager Image
makes sense and defines an analytic function of z in the following two cases [3] :
(i) 0 < │z│ < ∞ and
PPT Slide
Lager Image
(ii) μ1 = 0, 0 < │z│ τ˗1 and
PPT Slide
Lager Image
4. FUNCTIONAL RELATIONS INVOLVING GENERALIZED MELLIN-BARNES TYPE OF CONTOUR INTEGRAL
Here we give two interesting double summation formulas involving the
PPT Slide
Lager Image
-function asserted by the following theorem.
Theorem 4. If each of the series involved converges absolutely, the following formulas hold :
PPT Slide
Lager Image
and
PPT Slide
Lager Image
where C > 0, ℜ( μ + β ) > 0, ℜ( μ + η ) > 0,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
and ς ( h ; k ) are given in (3.4) and (3.5), respectively.
Proof . In view of the
PPT Slide
Lager Image
-function representation (3.3), we apply Theorem 2 by setting ρ ρ ( k ) = C ς ( h, k ) ( C > 0) and
PPT Slide
Lager Image
, where
PPT Slide
Lager Image
and ς ( h, k ) are defined by (3.4) and (3.5), respectively. Then we have
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
-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.
References
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