• K C Gupta

Articles written in Proceedings – Mathematical Sciences

• Unified theorems involvingH-function transform and Meijer Bessel function transform

A theorem which reveals an interesting relationship between originals of related functions inH-function transform and Meijer Bessel function transforms is established. Another theorem interconnecting the two Meijer Bessel function transforms is also obtained. These theorems seem to be capable of unifying and extending a large number of results obtained by earlier workers. Finally, we evaluate an integral by way of illustration.

• On the inverse Laplace transform of the product of a general class of polynomials and the multivariable H-function

In this paper we evaluate the inverse Laplace transform of$$\begin{gathered} s^{ - \eta } (s^{l_1 } + \lambda _1 )^{ - \sigma } (s^{l_2 } + \lambda _2 )^{ - \rho } \hfill \\ \times S_n^m [xs^{ - W} (S^{l_1 } + \lambda _1 )^{ - \upsilon } (S^{l_2 } + \lambda _2 )^{ - w} ]S_{n'}^{m'} [ys^{ - w'} (S^{l_1 } + \lambda _1 )^{ - \upsilon '} (S^{l_2 } + \lambda _2 )^{ - w_r } ] \hfill \\ \times H[z_1 s^{ - W_1 } (S^{l_1 } + \lambda _1 )^{ - \upsilon _1 } (S^{l_2 } + \lambda _2 )^{ - w_1 } ,...,z_r s^{ - w_r } (S^{l_1 } + \lambda _1 )^{ - \upsilon _r } (S^{l_2 } + \lambda _2 )^{ - w'} ] \hfill \\ \end{gathered}$$

Due to the general nature of the multivariable H-function involved herein, the inverse Laplace transform of the product of a large number of special functions involving one or more variables, occurring frequently in the problems of theoretical physics and engineering sciences can be obtained as simple special cases of our main findings. For the sake of illustration, we obtain here the inverse Laplace transform of a product of the Hermite polynomials, the Jacobi polynomials andr different modified Bessel functions of the second kind. A theorem obtained by Srivastava and Singh follows as a special case of our main result.

• On composition of some general fractional integral operators

In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman  and Erdélyi  follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.

• Convolution integral equations involving a general class of polynomials and the multivariableH-function

In this paper we first solve a convolution integral equation involving product of the general class of polynomials and theH-function of several variables. Due to general nature of the general class of polynomials and theH-function of several variables which occur as kernels in our main convolution integral equation, we can obtain from it solutions of a large number of convolution integral equations involving products of several useful polynomials and special functions as its special cases. We record here only one such special case which involves the product of general class of polynomials and Appell's functionF3. We also give exact references of two results recently obtained by Srivastavaet al  and Rashmi Jain  which follow as special cases of our main result.

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019