ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 4(2010), Pages 35-44.
ON GENERALIZED FRACTIONAL 𝑞-INTEGRAL OPERATORS INVOLVING THE 𝑞-GAUSS HYPERGEOMETRIC FUNCTION
(DEDICATED IN OCCASION OF THE 70-YEARS OF PROFESSOR HARI M. SRIVASTAVA)
SUNIL DUTT PUROHIT, RAJENDRA KUMAR YADAV
Abstract. In this paper, we introduce two generalized operators of fractional 𝑞-integration, which may be regarded as extensions of Riemann-Liouville, Weyl and Kober fractional𝑞-integral operators. Certain interesting connection the- orems involving these operators and𝑞-Mellin transform are also discussed.
1. Introduction
The fractional integration operators involving various special functions, in par- ticular the Gaussian hypergeometric functions, have found significant importance and applications in various sub-fields of applicable mathematical analysis. Since last three decades, a number of workers like Love [11], McBride [13], Kalla and Saxena [8, 9], Saigo [21-23], Saigo and Raina [24] etc. have studied in depth, the properties, applications and different extensions of various hypergeometric opera- tors of fractional integration. A detailed account of such operators along with their properties and applications can be found in the research monographs by Miller and Ross [14], Kiryakova [10] and Nishimoto [15-18] etc.
The fractional𝑞-calculus is the 𝑞-extension of the ordinary fractional calculus.
The theory of𝑞-calculus operators in recent past have been applied in the areas like ordinary fractional calculus, optimal control problems, solutions of the𝑞-difference (differential) and 𝑞-integral equations, 𝑞-transform analysis etc. Recently, Abu- Risha, Annaby, Ismail and Mansour [1] and Mansour [12] derived the fundamental set of solutions for the homogenous linear sequential fractional 𝑞-difference equa- tions with constant coefficients. Fang [6] and Purohit [19] deduced several trans- formations and summations formulae for the basic hypergeometric functions as the applications of fractional𝑞-differential operator. For more details one may refer the
2000Mathematics Subject Classification. 26A33, 33D15.
Key words and phrases. Saigo operators of fractional integration, Riemann-Liouville, Weyl and Kober fractional𝑞-integral operators,𝑞-Gauss hypergeometric function.
c
⃝2010 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted May 04, 2009. Published September 18, 2010.
35
recent papers [4], [5] and [20] on the subject.
We propose to define and investigate the𝑞-extensions of the hypergeometric op- erators of fractional integration due to Saigo [21].
In a series of papers [21-23], Saigo introduced the following pair of hypergeomet- ric operators of fractional integration.
For𝛼 >0, real numbers𝛽 and𝜂, we have:
𝐼0,𝑥𝛼,𝛽,𝜂𝑓(𝑥) =𝑥−𝛼−𝛽 Γ(𝛼)
∫ 𝑥 0
(𝑥−𝑡)𝛼−1 2𝐹1(𝛼+𝛽,−𝜂;𝛼; 1−𝑡/𝑥)𝑓(𝑡)𝑑𝑡, (1.1) 𝐽𝑥,∞𝛼,𝛽,𝜂𝑓(𝑥) = 1
Γ(𝛼)
∫ ∞ 𝑥
(𝑡−𝑥)𝛼−1 𝑡−𝛼−𝛽2𝐹1(𝛼+𝛽,−𝜂;𝛼; 1−𝑥/𝑡)𝑓(𝑡)𝑑𝑡, (1.2) where, the2𝐹1(.) function occurring in the right-hand side of the above equations, is the familiar Gaussian hypergeometric function defined as:
2𝐹1(𝑎, 𝑏;𝑐;𝑥)≡ 2𝐹1
⎡
⎣ 𝑎, 𝑏
; 𝑥 𝑐
⎤
⎦=
∞
∑
𝑛=0
(𝑎)𝑛(𝑏)𝑛
(𝑐)𝑛 𝑥𝑛
𝑛!. (1.3)
The operator𝐼0,𝑥𝛼,𝛽,𝜂(.) contains both the Riemann-Liouville and the Erd´𝑒lyi-Kober fractional integral operators, by means of the following relationships:
𝑅𝛼0,𝑥𝑓(𝑥) =𝐼0,𝑥𝛼,−𝛼,𝜂𝑓(𝑥) = 1 Γ(𝛼)
∫ 𝑥 0
(𝑥−𝑡)𝛼−1𝑓(𝑡)𝑑𝑡, (1.4) and
𝐸0,𝑥𝛼,𝜂𝑓(𝑥) =𝐼0,𝑥𝛼,0,𝜂𝑓(𝑥) =𝑥−𝛼−𝜂 Γ(𝛼)
∫ 𝑥 0
(𝑥−𝑡)𝛼−1 𝑡𝜂𝑓(𝑡)𝑑𝑡, (1.5) where as the operator (1.2) unifies the Weyl type and the Erd´𝑒lyi-Kober fractional integral operators. Indeed we have
𝑊𝑥,∞𝛼 𝑓(𝑥) =𝐽𝑥,∞𝛼,−𝛼,𝜂𝑓(𝑥) = 1 Γ(𝛼)
∫ ∞ 𝑥
(𝑡−𝑥)𝛼−1𝑓(𝑡)𝑑𝑡, (1.6) and
𝐾𝑥,∞𝛼,𝜂𝑓(𝑥) =𝐽𝑥,∞𝛼,0,𝜂𝑓(𝑥) = 𝑥𝜂 Γ(𝛼)
∫ ∞ 𝑥
(𝑡−𝑥)𝛼−1 𝑡−𝛼−𝜂𝑓(𝑡)𝑑𝑡. (1.7) For real or complex𝑎and∣𝑞∣<1, the𝑞-shifted factorial is defined as:
(𝑎;𝑞)0= 1, (𝑎;𝑞)𝑛=
𝑛−1
∏
𝑖=0
(1−𝑎𝑞𝑖), 𝑛 >0, 𝑎𝑛𝑑(𝑎;𝑞)∞=
∞
∏
𝑖=0
(1−𝑎𝑞𝑖). (1.8) Equivalently
(𝑎;𝑞)𝑛= Γ𝑞(𝑎+𝑛)(1−𝑞)𝑛
Γ𝑞(𝑎) , (1.9)
where the𝑞-gamma function cf. Gasper and Rahman [7], is given by Γ𝑞(𝑎) = (𝑞;𝑞)∞
(𝑞𝑎;𝑞)∞(1−𝑞)𝑎−1 = (𝑞;𝑞)𝑎−1
(1−𝑞)𝑎−1, (1.10) (𝑎∕= 0,−1,−2,⋅ ⋅ ⋅).
Also, the 𝑞-analogue of the power (binomial) function (𝑥+𝑦)𝑛 cf. Gasper and Rahman (see also Ernst [5]), is given by
(𝑥+𝑦)(𝑛)=
⎧
⎨
⎩
𝑥𝑛(−𝑥𝑦;𝑞)𝑛 , 𝑥∕= 0 𝑞𝑛(𝑛−1)/2 𝑦𝑛 , 𝑥= 0,
(1.11) where the𝑞-binomial coefficient is defined as:
[ 𝑛 𝑘
]
𝑞
= (𝑞;𝑞)𝑛
(𝑞;𝑞)𝑘 (𝑞;𝑞)𝑛−𝑘 . (1.12)
For a bounded sequence (𝐴𝑛)𝑛∈ℤof real or complex numbers, let𝑓(𝑥) =
+∞
∑
𝑛=−∞
𝐴𝑛𝑥𝑛
be a power series in𝑥, then the𝑞-translation operator is defined as:
𝒯𝑞,𝑦(𝑓(𝑥)) =
+∞
∑
𝑛=−∞
𝐴𝑛𝑥𝑛(𝑦/𝑥;𝑞)𝑛. (1.13) The generalized basic hypergeometric series cf. Gasper and Rahman [7] is given by
𝑟Φ𝑠
⎡
⎣
𝑎1,⋅ ⋅ ⋅ , 𝑎𝑟
; 𝑞, 𝑥 𝑏1,⋅ ⋅ ⋅, 𝑏𝑠
⎤
⎦=
∞
∑
𝑛=0
(𝑎1,⋅ ⋅ ⋅ , 𝑎𝑟;𝑞)𝑛 (𝑞, 𝑏1,⋅ ⋅ ⋅ , 𝑏𝑠;𝑞)𝑛
𝑥𝑛 {
(−1)𝑛𝑞𝑛(𝑛−1)/2}(1+𝑠−𝑟)
, (1.14) where
(𝑎1,⋅ ⋅ ⋅ , 𝑎𝑟;𝑞)𝑛= (𝑎1;𝑞)𝑛(𝑎2;𝑞)𝑛⋅ ⋅ ⋅(𝑎𝑟;𝑞)𝑛,
and for convergence, we have∣𝑞∣<1 and∣𝑥∣<1 if𝑟=𝑠+ 1, and for any𝑥if𝑟≤𝑠.
A𝑞-analogue of the familiar Riemann-Liouville fractional integral operator of a function𝑓(𝑥) due to Agarwal [2] is defined as:
𝐼𝑞𝛼{𝑓(𝑥)}= 𝑥𝛼−1 Γ𝑞(𝛼)
∫ 𝑥 0
(𝑞𝑡/𝑥;𝑞)𝛼−1𝑓(𝑡)𝑑𝑞𝑡, (1.15) whereℜ(𝛼)>0;∣𝑞∣<1 and
(𝑎;𝑞)𝛼= (𝑎;𝑞)∞
(𝑎𝑞𝛼;𝑞)∞, 𝛼∈ℝ.
Also, the basic analogue of the Kober fractional integral operator cf. Agarwal [2]
is defined by
𝐼𝑞𝜂,𝛼{𝑓(𝑥)}=𝑥−𝜂−1 Γ𝑞(𝛼)
∫ 𝑥 0
(𝑞𝑡/𝑥;𝑞)𝛼−1𝑡𝜂𝑓(𝑡)𝑑𝑞𝑡, (1.16) whereℜ(𝛼)>0;∣𝑞∣<1;𝜂∈ℝ.
A𝑞-analogue of the Weyl fractional integral operator (1.6) due to Al-Salam [3], is defined as:
𝐾𝑞𝛼𝑓(𝑥) = 𝑞−𝛼(𝛼−1)/2 Γ𝑞(𝛼)
∫ ∞ 𝑥
𝑡𝛼−1(𝑥/𝑡;𝑞)𝛼−1𝑓(𝑡𝑞1−𝛼)𝑑𝑞𝑡, (1.17) whereℜ(𝛼)>0;∣𝑞∣<1.
In the same paper, Al-Salam [3] introduced the𝑞-analogue of the operator (1.7) in
the following manner:
𝐾𝑞𝜂,𝛼{𝑓(𝑥)}=𝑞−𝜂𝑥𝜂 Γ𝑞(𝛼)
∫ ∞ 𝑥
(𝑥/𝑡;𝑞)𝛼−1𝑡−𝜂−1𝑓(𝑡𝑞1−𝛼)𝑑𝑞𝑡, (1.18) whereℜ(𝛼)>0;∣𝑞∣<1;𝜂∈ℝ.
Also the basic integrals (cf. Gasper and Rahman [7]), are defined as:
∫ 𝑥 0
𝑓(𝑡)𝑑𝑞𝑡=𝑥(1−𝑞)
∞
∑
𝑘=0
𝑞𝑘𝑓(𝑧𝑞𝑘), (1.19)
∫ ∞ 𝑥
𝑓(𝑡)𝑑𝑞𝑡=𝑥(1−𝑞)
∞
∑
𝑘=1
𝑞−𝑘𝑓(𝑥𝑞−𝑘), (1.20)
and
∫ ∞ 0
𝑓(𝑡)𝑑𝑞𝑡= (1−𝑞)
∞
∑
𝑘=−∞
𝑞𝑘𝑓(𝑞𝑘). (1.21)
The𝑞-binomial summation theorem is given by
1Φ0[𝑎;−;𝑞, 𝑧] = (𝑎𝑧;𝑞)∞ (𝑧;𝑞)∞
, ∣𝑧∣<1. (1.22)
Also the𝑞-Chu-Vondermonde summation theorem cf. Gasper and Rahman [7]
2Φ1
⎡
⎣
𝑞−𝑛, 𝑎
; 𝑞, 𝑞 𝑐
⎤
⎦= (𝑐/𝑎;𝑞)𝑛
(𝑐;𝑞)𝑛 (𝑎)𝑛. (1.23) The object of this paper is to introduce two hypergeometric operators of frac- tional𝑞-integration, which may be regarded as extensions of the fractional𝑞-integral operators (1.15)-(1.18). Having defined a𝑞-extensions of these operaotrs, we inves- tigate their fundamental properties such as integration by parts and connection theorems with𝑞-analogue of Mellin transform. Certain interesting special cases in the form of the known results have also been discussed.
2. The Fractional𝑞-Integral Operators
In this section, we introduce the following fractional 𝑞-integral operators in- volving the Gaussian basic hypergeometric function, which may be regarded as 𝑞-extensions of the Saigo operators (1.1) and (1.2).
For 𝛼 and real 𝛽, we define the fractional 𝑞-integral operators 𝐼𝑞𝛼,𝛽,𝜂(.) and 𝐾𝑞𝛼,𝛽,𝜂(.) in the following manner:
𝐼𝑞𝛼,𝛽,𝜂𝑓(𝑥) =𝑥−𝛽−1𝑞−𝜂(𝛼+𝛽) Γ𝑞(𝛼)
×
∫ 𝑥 0
(𝑡𝑞/𝑥;𝑞)𝛼−1 𝒯
𝑞,𝑞𝛼+1𝑥 𝑡
(
2Φ1
[𝑞𝛼+𝛽, 𝑞−𝜂;𝑞𝛼;𝑞, 𝑞])
𝑓(𝑡)𝑑𝑞𝑡, ∣𝑡/𝑥∣<1, (2.1) and
𝐾𝑞𝛼,𝛽,𝜂𝑓(𝑥) =𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−2𝛽
Γ𝑞(𝛼)
×
∫ ∞ 𝑥
(𝑥/𝑡;𝑞)𝛼−1𝑡−𝛽−1𝒯
𝑞,𝑞𝛼+1𝑡 𝑥
(
2Φ1[
𝑞𝛼+𝛽, 𝑞−𝜂;𝑞𝛼;𝑞, 𝑞])
𝑓(𝑡𝑞1−𝛼)𝑑𝑞𝑡,∣𝑥/𝑡∣<1, (2.2) where𝜂 is any non negative integer and the2Φ1(.) function occurring in the right- hand side of (2.1) and (2.2) is the Gaussian𝑞-hypergeometric function defined as special case (for𝑟= 2 and𝑠= 1) of the power series (1.14). Using series definitions of the basic integrals given by (1.19)-(1.20) and 𝑞-translation operator (1.13), we define the series representation for the operators (2.1) and (2.2) as:
𝐼𝑞𝛼,𝛽,𝜂𝑓(𝑥) =𝑥−𝛽𝑞−𝜂(𝛼+𝛽)(1−𝑞)𝛼
×
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛
𝑞𝑛
∞
∑
𝑘=0
𝑞𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘
𝑓(𝑥𝑞𝑘), (2.3) and
𝐾𝑞𝛼,𝛽,𝜂𝑓(𝑥) =𝑥−𝛽𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼
×
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛 (𝑞;𝑞)𝑛
𝑞𝑛
∞
∑
𝑘=0
𝑞𝛽𝑘(𝑞𝛼+𝑛;𝑞)𝑘 (𝑞;𝑞)𝑘
𝑓(𝑥𝑞−𝛼−𝑘), (2.4) where𝛼 >0,𝛽 being real number, and𝜂 is any non negative integer.
3. Fractional 𝑞-Integral Images of 𝑥𝜆−1
This section envisage the evaluation of the 𝑞-images of an elimentary function 𝑥𝜆−1 under the generalized fractional𝑞-integral operators introduced in the previ- ous section.
Theorem 1. If ∣𝑞∣<1,𝜆 >0 and(𝜆−𝛽+𝜂)>0, then 𝐼𝑞𝛼,𝛽,𝜂{
𝑥𝜆−1}
= Γ𝑞(𝜆)Γ𝑞(𝜆−𝛽+𝜂)
Γ𝑞(𝜆−𝛽)Γ𝑞(𝜆+𝛼+𝜂)𝑥𝜆−𝛽−1. (3.1) Proof. To prove the theorem (3.1), we take𝑓(𝑥) =𝑥𝜆−1in the series definition of fractional 𝑞-integral operator 𝐼𝑞𝛼,𝛽,𝜂(.), given by (2.3), the left-hand side yields to
𝐼𝑞𝛼,𝛽,𝜂{ 𝑥𝜆−1}
=𝑥𝜆−𝛽−1𝑞−𝜂(𝛼+𝛽)(1−𝑞)𝛼
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛 (𝑞;𝑞)𝑛
𝑞𝑛
×
∞
∑
𝑘=0
𝑞𝜆𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘
. (3.2)
On summing the inner1Φ0(.) series with the help of the equation (1.22), it reduces to
𝐼𝑞𝛼,𝛽,𝜂{ 𝑥𝜆−1}
=𝑥𝜆−𝛽−1𝑞−𝜂(𝛼+𝛽)(1−𝑞)𝛼
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛(𝑞𝜆;𝑞)𝛼+𝑛 𝑞𝑛, (3.3) on simplification and the usage of the 𝑞-Chu-Vondermonde summation theorem given by (1.23), the above equation leads to Theorem 1.
Theorem 2. If ∣𝑞∣<1,(𝛽−𝜆+ 1)>0 and(𝜂−𝜆+ 1)>0, then 𝐾𝑞𝛼,𝛽,𝜂{
𝑥𝜆−1}
= Γ𝑞(𝛽−𝜆+ 1)Γ𝑞(𝜂−𝜆+ 1)
Γ𝑞(1−𝜆)Γ𝑞(𝛽+𝛼−𝜆+𝜂+ 1)𝑥𝜆−𝛽−1𝑞𝛼(1−𝜆)−𝛼(𝛼+1)/2−𝛽. (3.4) Proof. On employing the definition (2.4) with𝑓(𝑥) =𝑥𝜆−1, we obtain
𝐾𝑞𝛼,𝛽,𝜂{ 𝑥𝜆−1}
=𝑥𝜆−𝛽−1𝑞𝛼(1−𝜆)−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼
×
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛 𝑞𝑛
∞
∑
𝑘=0
𝑞(𝛽−𝜆+1)𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘 . (3.5)
On summing the inner1Φ0(.) series with the help of the equation (1.22), it leads to 𝐾𝑞𝛼,𝛽,𝜂{
𝑥𝜆−1}
=𝑥𝜆−𝛽−1𝑞𝛼(1−𝜆)−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼
×
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛(𝑞𝛽−𝜆+1;𝑞)𝛼+𝑛
𝑞𝑛, (3.6)
which, on using the𝑞-Vondermonde summation theorem (1.23) and some simplifi- cations, leads to the proof of the result (3.4).
Further, it is interesting to observe that the newly defined operators (2.1) and (2.2) can be regarded as extensions of Riemann-Liouville, Weyl and Kober fractional 𝑞-integral operators with the following functional relations:
𝐼𝑞𝛼,0,𝜂𝑓(𝑥) =𝐼𝑞𝜂,𝛼𝑓(𝑥), (3.7) 𝐼𝑞𝛼,−𝛼,𝜂𝑓(𝑥) =𝐼𝑞𝛼𝑓(𝑥), (3.8) 𝐾𝑞𝛼,0,𝜂𝑓(𝑥) =𝑞−𝛼(𝛼+1)/2𝐾𝑞𝜂,𝛼𝑓(𝑥), (3.9) 𝐾𝑞𝛼,−𝛼,𝜂𝑓(𝑥) =𝐾𝑞𝛼𝑓(𝑥). (3.10) 4. Fractional Integration by Parts
In this section, we shall prove a theorem involving an important relationship between the operators𝐼𝑞𝛼,𝛽,𝜂(.) and𝐾𝑞𝛼,𝛽,𝜂(.):
Theorem 3. If 𝛼 > 0, 𝛽 a real number, and 𝜂 being a non negative integer, then
∫ ∞ 0
𝑓(𝑥)𝐾𝑞𝛼,𝛽,𝜂𝑔(𝑥)𝑑𝑞𝑡=𝑞−𝛼(𝛼+1)/2−𝛽∫ ∞ 0
𝑔(𝑥𝑞−𝛼)𝐼𝑞𝛼,𝛽,𝜂𝑓(𝑥)𝑑𝑞𝑡. (4.1) Provided that both of the𝑞-integrals exist.
Proof. On using the series definition of 𝑞-Saigo operator 𝐾𝑞𝛼,𝛽,𝜂(.), given by (2.4), the left-hand side, say𝐿of Equation (4.1) yields to
𝐿=
∫ ∞ 0
𝑓(𝑥)𝑥−𝛽𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛 (𝑞;𝑞)𝑛
𝑞𝑛
×
∞
∑
𝑘=0
𝑞𝛽𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘 𝑔(𝑥𝑞−𝛼−𝑘)𝑑𝑞𝑡. (4.2)
On changing the order of integration and summations in the above expression, which is valid under conditions mentioned with (2.4) and using the integral (1.21), the above equation reduces to
𝐿=𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼+1
∞
∑
𝑟=−∞
𝑞𝑟𝑓(𝑞𝑟)𝑞−𝑟𝛽+𝛽
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛
𝑞𝑛
×
∞
∑
𝑘=0
𝑞𝛽𝑘(𝑞𝛼+𝑛;𝑞)𝑘 (𝑞;𝑞)𝑘
𝑔(𝑞𝑟−𝛼−𝑘)
=𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽(1−𝑞)𝛼+1
∞
∑
𝑡=−∞
𝑞𝑡𝑔(𝑞𝑡−𝛼)𝑞−𝑡𝛽+𝛽
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛 𝑞𝑛
×
∞
∑
𝑘=0
𝑞𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘
𝑓(𝑞𝑡+𝑘), (4.3)
on replacing the basic bilateral series in the above relation by the integral (1.21), we obtain
𝐿=𝑞−𝜂(𝛼+𝛽)−𝛼(𝛼+1)/2−𝛽∫ ∞ 0
𝑔(𝑥𝑞−𝛼)𝑥−𝛽𝑞𝛽(1−𝑞)𝛼
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛 (𝑞;𝑞)𝑛
𝑞𝑛
×
∞
∑
𝑘=0
𝑞𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘 𝑓(𝑥𝑞𝑘)𝑑𝑞𝑡. (4.4) On interpreting the above expression in light of the series definition (2.3) of the𝑞- Saigo operator𝐼𝑞𝛼,𝛽,𝜂(.), the above equation (4.4) finally reduces to the right-hand side of the Theorem 3.
Interestingly, on setting𝛽 = 0 and employing the relations (3.7) and (3.9), the Theorem 3 yields to the following Corollary:
Corollary 1. For𝛼 >0and𝜂 being a non negative integer, the following result holds:
∫ ∞ 0
𝑓(𝑥)𝐾𝑞𝜂,𝛼𝑔(𝑥)𝑑𝑞𝑡=
∫ ∞ 0
𝑔(𝑥𝑞−𝛼)𝐼𝑞𝜂,𝛼𝑓(𝑥)𝑑𝑞𝑡. (4.5) Provided both of the 𝑞-integrals exist.
Further, if we replace𝛽 by−𝛼and make use of the relations (3.8) and (3.10), in the Theorem 3, we obtain yet another corollary providing interesting relationship between the operators𝐾𝑞𝛼(.) and𝐼𝑞𝛼(.) namely:
Corollary 2. For𝛼 >0and𝜂 being a non negative integer, the following result holds:
𝑞𝛼(𝛼−1)/2
∫ ∞ 0
𝑓(𝑥)𝐾𝑞𝛼𝑔(𝑥)𝑑𝑞𝑡=
∫ ∞ 0
𝑔(𝑥𝑞−𝛼)𝐼𝑞𝛼𝑓(𝑥)𝑑𝑞𝑡. (4.6) Provided that both of the𝑞-integrals exist.
Finally, it is worth mentioning that, if we remove the non negativity restriction on the parameter𝜂, the corollaries (4.5) and (4.6) reduces to the known results due to Agarwal [2].
5. The 𝑞-Mellin Transform of the 𝑞-Saigo Operators
In this section, we shall prove two theorems, which exhibit the connection be- tween the𝑞-Mellin transform and the operators given by Equations (2.1) and (2.2).
Theorem 4. If 𝛼 >0 and𝑠 <1 +𝑚𝑖𝑛{0, 𝜂−𝛽}, then 𝑀𝑞(
𝑥𝛽 𝐼𝑞𝛼,𝛽,𝜂𝑓(𝑥))
(𝑠) = Γ𝑞(1−𝑠)Γ𝑞(𝜂+ 1−𝑠−𝛽)
Γ𝑞(1−𝑠−𝛽)Γ𝑞(𝜂+ 1−𝑠+𝛼)𝑀𝑞(𝑓(𝑥)) (𝑠), (5.1) where the𝑞-Mellin transform of 𝑓(𝑥)is defined as:
𝑀𝑞(𝑓(𝑥)) (𝑠) = 1 (1−𝑞)
∫ ∞ 0
𝑥𝑠−1𝑓(𝑥)𝑑𝑞𝑡=
∞
∑
𝑟=−∞
𝑞𝑟𝑠𝑓(𝑞𝑟). (5.2) Proof. On using the definition (5.2) and the series definition of fractional 𝑞- integral operator𝐼𝑞𝛼,𝛽,𝜂(.) given by (2.3), the left-hand side (say𝐿) becomes
𝐿=
∞
∑
𝑟=−∞
𝑞𝑟𝑠−𝜂(𝛼+𝛽)(1−𝑞)𝛼
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛 𝑞𝑛
∞
∑
𝑘=0
𝑞𝑘(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘 𝑓(𝑞𝑟+𝑘)
=
∞
∑
𝑟=−∞
𝑞𝑟𝑠−𝜂(𝛼+𝛽)(1−𝑞)𝛼𝑓(𝑞𝑟)
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛 𝑞𝑛
∞
∑
𝑘=0
𝑞𝑘(1−𝑠)(𝑞𝛼+𝑛;𝑞)𝑘
(𝑞;𝑞)𝑘 . (5.3) On summing the inner1Φ0(.) series with the help of the equation (1.22), it reduces to
𝐿=
∞
∑
𝑟=−∞
𝑞𝑟𝑠−𝜂(𝛼+𝛽)(1−𝑞)𝛼𝑓(𝑞𝑟)
𝜂
∑
𝑛=0
(𝑞𝛼+𝛽;𝑞)𝑛(𝑞−𝜂;𝑞)𝑛
(𝑞;𝑞)𝑛(𝑞1−𝑠;𝑞)𝛼+𝑛𝑞𝑛, (5.4) which further simplifies to
𝐿= Γ𝑞(1−𝑠)Γ𝑞(𝜂+ 1−𝑠−𝛽) Γ𝑞(1−𝑠−𝛽)Γ𝑞(𝜂+ 1−𝑠+𝛼)
∞
∑
𝑟=−∞
𝑞𝑟𝑠𝑓(𝑞𝑟). (5.5) On interpreting the basic bilateral series in light of the definition (5.2), the above equation yields to the right-hand side of the theorem (5.1).
Theorem 5. If 𝛼 >0 and𝑠 >−𝑚𝑖𝑛{𝛽, 𝜂}, then following relation holds:
𝑀𝑞
(𝑥𝛽 𝐾𝑞𝛼,𝛽,𝜂𝑓(𝑥))
(𝑠) = Γ𝑞(𝛽+𝑠)Γ𝑞(𝜂+𝑠)
Γ𝑞(𝑠)Γ𝑞(𝑠+𝛼+𝛽+𝜂)𝑞−𝛼(𝛼+1)/2−𝛽𝑀𝑞
(𝑓(𝑥𝑞−𝛼)) (𝑠), (5.6) where the𝑞-Mellin transform of 𝑓(𝑥)is given by the relation (5.2).
The proof of the above theorem follows similarly.
If we set𝛽= 0 and make use of relations (3.7) and (3.9), the results of Theorems 4 and 5 respectively give rise to the following corollaries involving relations between the𝑞-Mellin transform and the Kober fractional𝑞-integral operators:
Corollary 3. If 𝛼 >0 and(1−𝑠)>0, then 𝑀𝑞
(𝐼𝑞𝜂,𝛼𝑓(𝑥))
(𝑠) = Γ𝑞(𝜂+ 1−𝑠)
Γ𝑞(𝜂+ 1−𝑠+𝛼)𝑀𝑞(𝑓(𝑥)) (𝑠), (5.7) and
Corollary 4. If 𝛼 >0 and(𝜂+𝑠)>0, then following relation holds:
𝑀𝑞(
𝐾𝑞𝜂,𝛼𝑓(𝑥))
(𝑠) = Γ𝑞(𝜂+𝑠) Γ𝑞(𝜂+𝑠+𝛼)𝑀𝑞(
𝑓(𝑥𝑞−𝛼))
(𝑠). (5.8)
Finaly, if we replace 𝛽 by −𝛼and make use of the relations (3.8) and (3.10), Theorems 4 and 5 yield the following corollaries:
Corollary 5. For𝛼 >0 and(1−𝑠)>0, following result holds:
𝑀𝑞(
𝑥−𝛼 𝐼𝑞𝛼𝑓(𝑥))
(𝑠) = Γ𝑞(1−𝑠)
Γ𝑞(1−𝑠+𝛼)𝑀𝑞(𝑓(𝑥)) (𝑠), (5.9) and
Corollary 6. If 𝛼 >0 and(𝑠−𝛼)>0, then:
𝑀𝑞
(𝑥−𝛼𝐾𝑞𝛼𝑓(𝑥))
(𝑠) =Γ𝑞(𝑠−𝛼)
Γ𝑞(𝑠) 𝑞−𝛼(𝛼−1)/2 𝑀𝑞
(𝑓(𝑥𝑞−𝛼))
(𝑠). (5.10) 6. Concluding Observations
We briefly consider now some consequences of the results derived in the preceed- ing sections.
(i) If we let𝑞→1−, and make use of the limit formulae:
lim
𝑞→1−Γ𝑞(𝑎) = Γ(𝑎)𝑎𝑛𝑑 lim
𝑞→1−
(𝑞𝑎;𝑞)𝑛
(1−𝑞)𝑛 = (𝑎)𝑛 , (6.1) where
(𝑎)𝑛=𝑎(𝑎+ 1)⋅ ⋅ ⋅(𝑎+𝑛−1), (6.2) we observe that the operators (2.1) and (2.2) provides respectively, the𝑞-extensions of the known hypergeometric operators (1.1) and (1.2) due to Saigo [21].
(ii) Further, it is interesting to observe that the results given by (5.1) and (5.6) are the 𝑞-extensions of the known results due to Saigo, Saxena and Ram [25, pp.
295-296, eqn. (4.1) and (4.3)].
Acknowledgments. The authors would like to thank the anonymous referee for his/her comments that helped us improve this article.
References
[1] M.H. Abu-Risha, M.H. Annaby, M.E.H. Ismail and Z.S. Mansour,Linear𝑞-difference equa- tions,Z. Anal. Anwend.26(2007), 481-494.
[2] R.P. Agarwal, Certain fractional𝑞-integrals and 𝑞-derivatives,Proc. Camb. Phil. Soc.,66 (1969), 365-370.
[3] W.A. Al-Salam, Some fractional𝑞-integrals and 𝑞-derivatives,Proc. Edin. Math. Soc.,15 (1966), 135-140.
[4] G. Bangerezako, Variational calculus on𝑞-nonuniform lattices,J. Math. Anal. Appl. 306 (1)(2005), 161-179.
[5] T. Ernst,A method for𝑞-calculus,J. Nonlinear Math. Phys.,10(4)(2003), 487-525.
[6] J.P. Fang,Some applications of𝑞-differential operator,J. Korean Math. Soc.,47(2)(2010), 223-233.
[7] G. Gasper and M. Rahman,Basic Hypergeometric Series,Cambridge University Press, Cam- bridge, 1990.
[8] S.L. Kalla and R.K. Saxena,Integral operators involving hypergeometric functions,Math. Z., 108(1969), 231-234.
[9] S.L. Kalla and R.K. Saxena,Integral operators involving hypergeometric functions -II,Univ.
Nac. Tucum´𝑎n, Rev. Ser.,A24(1974), 31-36.
[10] V. Kiryakova,Generalized Fractional Calculus and Applications,Longman Scientific & Tech., Essex, 1994.
[11] E.R. Love,Some integral equations involving hypergeometric functions, Proc. Edin. Math.
Soc.,15(3)(1967), 169-198.
[12] Z.S.I. Mansour,Linear sequential𝑞-difference equations of fractional order, Fract. Calc. Appl.
Anal.,12(2)(2009), 159-178.
[13] A.C. McBride,Fractional powers of a class of ordinary differential operators,Proc. London Math. Soc. (III),45(1982), 519-546.
[14] K.S. Miller and B. Ross,An Introduction to the Fractional Calculus and Differential Equa- tions,A Wiley Interscience Publication. John Wiley and Sons Inc., New York, 1993.
[15] K. Nishimoto,Fractional Calculus,Descartes Press, Koriyama (Japan), 1984.
[16] K. Nishimoto,Fractional Calculus-II,Descartes Press, Koriyama (Japan), 1987.
[17] K. Nishimoto,Fractional Calculus-III,Descartes Press, Koriyama (Japan), 1989.
[18] K. Nishimoto,Fractional Calculus-IV,Descartes Press, Koriyama (Japan), 1991.
[19] S.D. Purohit, Summation formulae for basic hypergeometric functions via fractional 𝑞- calculus,Le Matematiche,64(1)(2009), 67-75.
[20] P.M. Rajkovi´𝑐, S.D. Marinkovi´𝑐, M.S. Stankovi´𝑐,Frcational integrals and derivatives in𝑞- calculus,Applicable Analysis and Discrete Mathematics,1(2007), 1-13.
[21] M. Saigo,A remark on integral operators involving the Guass hypergeometric function, Rep.
College General Ed., Kyushu Univ.,11(1978), 135-143.
[22] M. Saigo,A certain boundary value problem for the Euler-Darboux equation-I,Math. Japon- ica,24(4)(1979), 377-385.
[23] M. Saigo,A certain boundary value problem for the Euler-Darboux equation-II,Math. Japon- ica,25(2)(1980), 211-220.
[24] M. Saigo and R.K. Raina,On the fractional calculus operator involving Gauss’s series and its applications to certain statistical distortions,Rev. Tech. Ing. Univ. Zulia,14(1)(1991), 53-62.
[25] M. Saigo, R.K. Saxena and J. Ram,Certain properties of operators of fractional integration associated with Mellin and Laplace transformations, Current Topics in Analytic Function Theory, (1993), 291-304.
Sunil Dutt Purohit
Department of Basic-Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur-313001, India.
E-mail address:sunil a [email protected]
RAJENDRA KUMAR YADAV
Department of Mathematics and Statistics, J. N. Vyas University, Jodhpur-342005, In- dia.
E-mail address:[email protected]
