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Certain integrals involving multivariate Mittag-Leffler function
Journal of Inequalities and Applications volume 2019, Article number: 208 (2019)
Abstract
The objective of this article is to present several new integral equalities involving the multivariate Mittag-Leffler functions which are associated with the Laguerre polynomials. To emphasize our main results, we also consider some important special cases. The main results of our paper are quite general in nature and yield a very large number of integral equalities involving polynomials occurring in problems of mathematical analysis and mathematical physics.
1 Introduction and preliminaries
The function defined by the series representation
and its generalization
were introduced and studied by Agarwal [1], Mittag-Leffler [2, 3], Humburt [4], Humbert and Agrawal [5], and Wiman [6, 7], where \(\mathbb{C}\) is the set of complex numbers. The main properties of these functions are given in the book by Erdélyi et al. [8, Sect. 18.1], and a more extensive and detailed account on Mittag-Leffler functions is presented in Dzherbashyan [9, Chap. 2]. In particular, the functions (1) and (2) are entire functions of order \(\rho = 1/\xi \) and type \(\sigma = 1\); see, for example, [9, p. 118]. For a detailed account of various properties, generalizations, and applications of these functions, the reader may refer to an excellent work of Dzherbashyan [9], Kilbas and Saigo [10,11,12,13], Gorenflo and Mainardi [14], Gorenflo, Luchko and Rogosin [15], and Gorenflo, Kilbas and Rogosin [16].
The series representation of a generalization of (2) was introduced by Prabhaker [17] as:
where \(\xi, \nu, \delta \in \mathbb{C}\ (\Re (\xi ) > 0 )\). It is entire function of order \([\Re (\xi ) ] ^{-1} \) (see [17, p. 7]) and \((\delta )_{n} \) denotes the Pochhammer symbol defined as:
Srivastava and Tomoviski [18] studied and generalized the Mittag-Leffler-type function \(E_{\xi,\nu }^{\delta } (z)\) as
where \(\delta,\kappa,\xi,\nu,z\in \mathbb{C};\Re (\delta )>0, \Re (\kappa )>0,\Re (\xi )>0\).
A generalization of (5) was initiated by Salim and Faraj [19] as follows:
where \(\delta,\xi,\nu,\kappa,z\in \mathbb{C};\min (\Re (\delta ),\Re (\xi ),\Re (\nu ), \Re (\kappa )>0 );\rho,\varepsilon >0,\rho \le \Re (\xi )+\varepsilon \).
Further, a multivariate generalization of Mittag-Leffler function \(E_{\xi _{i};\nu; \varepsilon _{i} }^{\delta _{i};\kappa _{i};\rho _{i} } (\cdot )\), which is a generalization of (6), was studied by Gujar et al. [20] in the following form:
where \(\delta _{i},\xi _{i},\nu,\kappa _{i},z_{i}\in \mathbb{C}; \mathop{\min }_{1\le i\le r} (\Re (\delta _{i} ), \Re (\xi _{i} ),\Re (\nu ),\Re (\kappa _{i} ) )>0\) and \(\rho _{i},\varepsilon _{i} >0;\rho _{i} < \Re (\xi _{i} )+\varepsilon _{i}\ (i=1,2,\dots,r)\).
For a more detailed account of various properties, generalizations, and applications in terms of fractions of this function, the reader may refer to Mishra et al. [21], Purohit et al. [22], Saxena et al. [23] and Suthar et al. [24,25,26].
The polynomial \(L_{n}^{ (\mu,\tau )} (x )\) was defined by Prabhaker and Suman [27] as:
where \(\mu \in \mathbb{C}^{+},\tau \in \mathbb{C}_{-1}^{+} \); \(n\in \mathbb{N}\). If \(\mu =1\), then (8) reduces to
where \(L_{n}^{\tau } (x )\) is a well-known generalized Laguerre polynomial (see [28]).
In the sequel, the Konhauser polynomials of the second kind were defined by Srivastava [29] as:
where \(\tau \in \mathbb{C}_{-1}^{+} \), \(n\in \mathbb{N}\) and \(r\in \mathbb{Z}\).
It can be easily verified that
Further, the polynomial \(\mathrm{Z} _{n}^{ (\mu,\tau )} [x;r ]\) is defined [30] as:
From Eqs. (10) and (13), we obtain
If \(\mu \in {\mathrm{\mathbb{N}}} \), then Eq. (12) can be written in the following form:
The set of polynomials \(L_{n}^{ (\mu,\tau )} [\vartheta;x ]\) is defined [30] as:
where \(\mu,\vartheta \in \mathbb{C}^{+},\tau \in \mathbb{C}_{-1} ^{+} \), \(n\in \mathbb{N}\).
2 Main integral equalities
Throughout this paper, we assume that \(\delta _{i},\xi _{i},v,\kappa _{i},\alpha \in \mathbb{C}; \mathop{\min }_{1\le i\le r} (\Re (\delta _{i} ),\Re (\xi _{i} ), \Re (v ), \Re (\kappa _{i} ) )>0\), \((\rho _{i},\varepsilon _{i} )>0\) and \(\rho _{i} <\Re (\xi _{i} )+\varepsilon _{i} \); \(i=1,2, \dots,r\).
Theorem 1
The following integral equality holds:
Proof
Applying Eq. (7) to the left-hand side of Eq. (17), we obtain
This completes the proof of Theorem 1. □
Theorem 2
The following integral equality holds:
Proof
Using Eq. (6) in the left-hand side of Eq. (18), we obtain
Changing the variable s to \(u=\frac{s-t}{x-t} \), we obtain
from which, after little simplification, we easily arrive at the required Eq. (18). □
Theorem 3
If \(\omega _{i},\tau,\in \mathbb{C}; \mathop{\min }_{1\le i \le r} (\Re (\omega _{i} ),\Re (\tau ) )>0\), then the following integral equality holds:
Proof
Applying Eq. (7) to the left-hand side of Eq. (19), we obtain
Substituting \(\rho _{1} =\cdots =\rho _{r} =1\) in Eq. (20) reduces it to
Now applying the formula , Eq. (21) is reduced to the following form:
 □
Theorem 4
The following integral equality holds:
Proof
Applying Eq. (6) to the left-hand side of Eq. (22), we obtain
This completes the proof of Theorem 4. □
Remark 2.1
Upon setting \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots =\varepsilon _{r} =1\) and \(r=1\), Eqs. (17), (18), (19), and (22) reduce to a result given by Shukla and Prajapati [31, Eqs. (2.4.1), (2.4.2), (2.4.3), (2.4.4)].
Remark 2.2
By setting the parameters in Eqs. (17), (18), (19), and (22), we obtain the known results established by Khan [32, Eqs. (2.4.4), (2.4.5), (2.4.6), (2.4.7)].
Lemma 2.1
([33, Eq. 2.13])
If \(\mu _{1},\mu _{2}, \alpha ', \beta '\in \mathbb{C}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following formula holds:
Theorem 5
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where
Proof
Using Eqs. (6) and (23) in the left-hand side of Eq. (24), we obtain
Using the fact that , where \((-x )_{n} = (-1 )^{n} (x-n+1 ) _{n} \), on the right-hand side of Eq. (26), yields
from which, after little rearrangement, we easily arrive at the required Eq. (24). □
Theorem 6
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N,}}} \) then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Theorem 7
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Theorem 8
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}; \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Proof
The proofs of Eqs. (27), (28), and (29) are the same as those of Eq. (25), which can be obtained from Eqs. (18), (19), and (22). □
Theorem 9
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where
Proof
Using \((-x )_{n} = (-1 )^{n} (x-n+1 ) _{n} \), Eq. (26) is reduced to
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31). □
Theorem 10
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Theorem 11
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Theorem 12
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Proof
The proofs of Eqs. (32), (33), and (34) are the same as that of Eq. (30). □
Remark 2.3
Upon setting \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots \varepsilon _{r} =1\), Eqs. (17), (18), (19), (22), (24), (27), (28), (29), (30), (32), (33), and (34) are reduced to Eqs. (2.1), (2.3), (2.5), (2.10), (2.20), (2.27), (2.28), (2.29), (2.31), (2.35), (2.36), and (2.37) established by Agarwal et al. [33].
3 Special cases
In this section, we emphasize special cases by selecting particular values of parameters.
(i) Putting \(\alpha '=\beta '=1\), the results in Eqs. (30), (32), (33), and (34) are reduced to the following form:
Corollary 1
If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 2
If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 3
If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \),; \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 4
If \(\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then
(ii) Putting \(\mu _{1} =\mu _{2} =\alpha '=\beta '=1\) and using Eq. (12), the results in Eqs. (30), (32), (33), and (34) reduced to the following form:
Corollary 5
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 6
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 7
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 8
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \), \(n,m\in {\mathrm{\mathbb{N}}} \), then
(iii) Putting \(\mu _{1} =\mu _{1} =0, \alpha '=\beta '=1\), the results in Eqs. (30), (32), (33), and (34) are reduced to the following form:
Corollary 9
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 10
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 11
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Corollary 12
If \(\eta \in \mathbb{C}^{+}, \tau _{1},\tau _{2} \in \mathbb{C}_{-1} ^{+} \) and \(n,m\in {\mathrm{\mathbb{N}}} \), then
Remark 3.1
If we put \(\kappa _{1} =\cdots =\kappa _{r} =\varepsilon _{1} =\cdots = \varepsilon _{r} =1\), then Eqs. (35)–(46) are reduced to Eqs. (3.1), (3.2), (3.3), (3.4), (3.6), (3.7), (3.8), (3.9), (3.11), (3.16), (3.17), (3.18) established in Agarwal et al. [33].
(iv) Setting \(\rho _{i} =\kappa _{i} =\varepsilon _{i} =1, \xi _{1} =\cdots =\xi _{r} =1\), the multivariate Mittag-Leffler function reduces to the confluent hypergeometric series, and we have the following results, which are obtained from the main results in (17), (18), (19), (22), (24), (27), (28), (29), (30), (32), (33), and (34).
Corollary 13
The following integral equality holds:
Corollary 14
Then following integral equality holds:
Corollary 15
Then following integral equality holds:
Corollary 16
Then following integral equality holds:
Corollary 17
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Corollary 18
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Corollary 19
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Corollary 20
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\Im _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (25).
Corollary 21
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Corollary 22
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \) and \(n,m\in {\mathrm{ \mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Corollary 23
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
Corollary 24
If \(\alpha ',\beta ',\eta \in \mathbb{C}^{+}, \mu _{1},\mu _{2},\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \), \(\tau _{1},\tau _{2} \in \mathbb{C}_{-1}^{+} \); \(n,m\in {\mathrm{\mathbb{N}}} \), then the following integral equality holds:
where \(\wp _{\mu _{1},\tau _{1},\mu _{2},\tau _{2} }^{m,n,\alpha ', \beta '} \) is given by Eq. (31).
(v) Setting \(\rho _{i} =\kappa _{i} =\varepsilon _{i} =\delta _{i} =\xi _{i} =v=r=1\), the multivariate Mittag-Leffler function is reduced to the exponential function \(E_{1;1; 1}^{1;1;1} (z )=E_{1,1} (z )=\exp (z)\); and we can find a similar line of results associated with exponential function from Eqs. (47)–(58).
References
Agarwal, R.P.: A propos d’une note de M. Pierre Humbert (French). C. R. Acad. Sci. 236, 2031–2032 (1953)
Mittag-Leffler, G.M.: Sur la nouvelle fonction \(E_{\alpha } (x)\). C. R. Math. Acad. Sci. Paris 137, 554–558 (1903)
Mittag-Leffler, G.M.: Sur la representation analytiqie d’une fonction monogene (cinquieme note). Acta Math. 29, 101–181 (1905)
Humbert, P.: Quelques resultants d’le function de Mittag-Leffler. C.R. Acad. Sci. Paris 236, 1467–1468 (1953)
Humbert, P., Agarwal, R.P.: Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations. Bull. Sci. Math. 77(2), 180–185 (1953)
Wiman, A.: Über den Fundamental satz in der Theorie der Functionen \(E_{\alpha } (x)\). Acta Math. 29, 191–201 (1905)
Wiman, A.: Über die Nullstellun der Funktionen \(E_{\alpha } (x)\). Acta Math. 29, 217–234 (1905)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, Vol. III. McGraw-Hill, New York (1955)
Dzherbashyan, M.M.: Integral Transforms and Representations of Function in Computer Domain (Russian). Nauka, Moscow (1966)
Kilbas, A.A., Saigo, M.: On solution of integral equations of Abel–Volterra type. Differ. Integral Equ. 8, 993–1011 (1995)
Kilbas, A.A., Saigo, M.: Fractional integrals and derivatives of Mittag-Leffler type function (Russian). Dokl. Akad. Nauk Belarusi 39(4), 22–26 (1995)
Kilbas, A.A., Saigo, M.: On Mittag-Leffler type function, fractional calculus operators and solutions of integral equations. Integral Transforms Spec. Funct. 4, 355–370 (1996)
Kilbas, A.A., Saigo, M.: Solution in closed form of a class of linear differential equations of fractional order. Differ. Equ. 33, 194–204 (1997)
Gorenflo, R., Mainardi, F.: The Mittag-Leffler type functions in the Riemann–Liouville fractional calculus. In: Boundary Value Problems. Special Functions and Fractional Calculus (Proc. Int. Conf. Minsk 1996), Belarusian State Univ., Minsk, pp. 215–225 (1996)
Gorenflo, R., Luchko, Y., Rogosin, S.V.: Mittag-Leffer type function, notes on growth properties and distribution of zeros. Preprint No. A04-97, Freie Universität Berlin, Serie A Mathematik, Berlin (1997)
Gorenflo, R., Kilbas, A.A., Rogosin, S.V.: On the generalized Mittag-Leffler type function. Integral Transforms Spec. Funct. 7, 215–224 (1998)
Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)
Srivastava, H.M., Tomovski, Z.: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 211(1), 198–210 (2009)
Salim, T.O., Faraj, A.W.: A generalization of Mittag-Leffler function and integral operator associated with fractional calculus. Fract. Calc. Appl. Anal. 3, 1–13 (2012)
Gurjar, M.K., Prajapati, J.C., Gupta, K.: A study of generalized Mittag-Leffler function via fractional calculus. J. Inequal. Spec. Funct. 5(3), 6–13 (2014)
Mishra, V.N., Suthar, D.L., Purohit, S.D.: Marichev–Saigo–Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function. Cogent Math. 4, Article ID 1320830 (2017)
Purohit, S.D., Kalla, S.L., Suthar, D.L.: Fractional integral operators and the multiindex Mittag-Leffler functions. Scientia, Ser. A, Math. Sci. 21, 87–96 (2011)
Saxena, R.K., Ram, J., Suthar, D.L.: Generalized fractional calculus of the generalized Mittag-Leffler functions. J. Indian Acad. Math. 31(1), 165–172 (2009)
Suthar, D.L., Amsalu, H.: Generalized fractional integral operators involving Mittag-Leffler function. Appl. Appl. Math. 12(2), 1002–1016 (2017)
Suthar, D.L., Habenom, H., Tadesse, H.: Generalized fractional calculus formulas for a product of Mittag-Leffler function and multivariable polynomials. Int. J. Appl. Comput. Math. 4(1), 1–12 (2018)
Suthar, D.L., Purohit, S.D.: Unified fractional integral formulae for the generalized Mittag-Leffler functions. J. Sci. Arts 27(2), 117–124 (2014)
Prabhakar, T.R., Suman, R.: Some results on the polynomials \(L_{n}^{\alpha,\beta } (x)\). Rocky Mt. J. Math. 8(4), 751–754 (1978)
Rainville, E.D.: Special Functions. Macmillan, New York (1960)
Srivastava, H.M.: A multilinear generating function for the Konhauser sets of bi-orthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math. 117(1), 183–191 (1985)
Shukla, A.K., Prajapati, J.C., Salehbhai, I.A.: On a set of polynomials suggested by the family of Konhauser polynomial. Int. J. Math. Anal. 3(13–16), 637–643 (2009)
Shukla, A.K., Prajapati, J.C.: On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336(2), 797–811 (2007)
Khan, M.A.: On some properties of the generalized Mittag-Leffler function. SpringerPlus 2, 337 (2013) https://doi.org/10.1186/2193-1801-2-337
Agarwal, P., Chand, M., Jain, S.: Certain integrals involving generalized Mittag-Leffler functions. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 85(3), 359–371 (2015)
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Suthar, D.L., Amsalu, H. & Godifey, K. Certain integrals involving multivariate Mittag-Leffler function. J Inequal Appl 2019, 208 (2019). https://doi.org/10.1186/s13660-019-2162-z
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DOI: https://doi.org/10.1186/s13660-019-2162-z