Some new results for the Srivastava-Luo-Raina $\mathbb{M}$-transform pertaining to the incomplete H-functions

1.   Introduction

There are numerous integral transforms available in the literature such as the Laplace transform, Mellin transform, Z-transform, Fourier transform and many more ^[1,2]. The integral transforms are very useful in solving differential equations of integer as well as fractional orders arising in scientific problems. In this sequence, recently, Srivastava et al. [3,p.1386,Eq.(1.1)] introduced the following $\mathbb{M}$-Transform:

where both $p \in \mathbb{C}$ and $q \in \mathbb{R_{+}}$ are the transform variables. The classical Stieltjes transform ^[4] is a special case of the Srivastava-Luo-Raina transform (1.1) when $p = 0$.

Convergence conditions of the integral transform (1.1) is given in the theorem [3,p. 1388].

The incomplete $H$-functions (IHF) $\gamma^{m, n}_{p, q}(w)$ and $\Gamma^{m, n}_{p, q}(w)$ have introduced and investigated by Srivastava et al. [5,Eqs.(2.1)-(2.4)] in the following manner:

where

and

where

The IHF $\gamma^{m, n}_{p, q}(z)$ and $\Gamma^{m, n}_{p, q}(z)$ in (1.2) and (1.4) exist for all $x \geqq 0$ under the same contour and the same set of conditions as stated in ^[5] (see, also, for details, ^[6,7,8,9]). A complete description of (1.2) and (1.4) can be found in ^[5].

Some interesting and important special cases of incomplete $H-$ functions are given below (see also ^[10]):

$\textbf{(i)}$ The case $u = 0$ of (1.4) reduces to the Fox's $H$-function

$\textbf{(ii)}$ Letting $m = 1$, $n = p$, $q$ being replaced by $q + 1$ and taking suitable parameters, the functions (1.2) and (1.4) convert, respectively, to the incomplete Fox-Wright functions(IFWF) $_p\Psi^{(\gamma)}_{q}$ and $_p\Psi^{(\Gamma)}_{q}$ (see [5,Eqs. (6.3) and (6.4)]):

and

$\textbf{(iii)}$ Taking $x = 0$, $m = n = p = 1$, $q$ being replaced by $m + 1$ and choosing proper parameters, the function (1.4) reduces to the generalized multi-index Mittag-Leffler function (GMIMLF) (see, details, ^[11])

In this paper, we aim to establish the images of incomplete H-functions and then, it is reduces to the some useful interesting special functions under this transform.

2.   The Srivastava-Luo-Raina $\mathbb{M}$-transform of IHF

In this section, we establish the Srivastava-Luo-Raina $\mathbb{M}$-transform of the incomplete H-functions (1.2) and (1.3).

Theorem 1. If $\mathfrak{R}(\lambda)\geq 0$, $\mu \in \mathbb{Z_{+}}$, $p \in \mathbb{C}$ and $q \in \mathbb{R_{+}}$ and with the conditions presented along with (1.2), the following $\mathbb{M}$-transform result exits for the IHF $\gamma^{M, N}_{P, Q}[z]$:

In the above equation $\mathbb{B}(x, y)$ indicates the classical Euler-Beta function.

Proof. To prove the result (2.1), by applying the Srivastava-Luo-Raina $\mathbb{M}$-transform given in (1.1) of (1.2), then interchanging the order of integral and contour integral (which is allowable under the conditions presented) and with the help of [3,p. 1389,Eq. (2.7)], we easily achieve the right hand side of (2.1) after a small adjustment of terms.

Theorem 2. If $\mathfrak{R}(\lambda)\geq 0$, $\mu \in \mathbb{Z_{+}}$, $p \in \mathbb{C}$ and $q \in \mathbb{R_{+}}$ and with the conditions presented along with (1.3), the following Srivastava-Luo-Raina $\mathbb{M}$-transform result exits for the IHF $\Gamma^{M, N}_{P, Q}[z]$:

In the above equation $\mathbb{B}(x, y)$ is the famous Euler-Beta function.

Proof. Similarly as in the proof of Theorem 1, one can easily derive the result. So, we omit the details.

Among many particular cases of the results in Theorems 1 and 2, we give some of them in the subsequent corollaries.

Corollary 2.1. If $\mathfrak{R}(\lambda)\geq 0$, $\mu \in \mathbb{Z_{+}}$, $p \in \mathbb{C}$ and $q \in \mathbb{R_{+}}$, the following $\mathbb{M}$-transform of the IFWF ${}_P\Psi^{(\gamma)}_{Q}[z]$ and ${}_P\Psi^{(\Gamma)}_{Q}[z]$ and MIMLF $E^{\rho, \sigma}_{(A_{j}, B_{j})_{M}}[z]$ exist:

and

and

here $\mathbb{B}(x, y)$ is the standards Euler-Beta function and provided both sided members of above identities exist.

Proof. Considering (1.7), (1.8) and (1.9), we can obtain the results here from those in Theorems 1 and 2.

3.   Conclusions

In this work, we have established image of incomplete $H$-functions in the Srivastava-Luo-Raina $\mathbb{M}$-transform and obtained the images of some useful and important cases of incomplete $H$-function in it. The results presented in this article are new and and can be utilized to derivative various new and known results having applications in science and engineering.

Acknowledgments

The author K. S. Nisar thanks to Prince Sattam bin Abdulaziz University, Saudi Arabia for providing facilities and support.

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.