New reduction formulas for the Humbert functions (the confluent Appell functions) ${\mathrm{\Psi }}_2$, ${\mathrm{\Phi }}_2$ and ${\mathrm{\Phi }}_3$ are given.

By employing two well-known Euler’s transformations for the hypergeometric function ${}_2{F}_1$, Liu and Wang established numerous general transformation and reduction formulas for the Kampé de Fériet function and deduced many new summation formulas for the Kampé de Fériet function with the aid of classical summation theorems for the ${}_2{F}_1$ due to Kummer, Gauss and Bailey. Here, by making a fundamental use of the above-mentioned reduction formulas, we aim to establish 32 general summation formulas for the Kampé de Fériet function with the help of generalizations of the above-referred summation formulas for the ${}_2{F}_1$ due to Kummer, Gauss and Bailey. Relevant connections of some particular cases of our main identities, among numerous ones, with those known formulas are explicitly indicated.

In the theory of generalized hypergeometric functions, classical summation theorems for the series ${}_2{F}_1$ , ${}_3{F}_2$ , ${}_4{F}_3$ , ${}_5{F}_4$ and ${}_7{F}_6$ play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions ${}_p{F}_p$ for $p=2,3,4,5$ and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings.

In this paper, we consider the matrix expression of convolution, and its generalized continuous form. The matrix expression of convolution is effectively applied in convolutional neural networks, and in this study, we correlate the concept of convolution in mathematics to that in convolutional neural network. Of course, convolution is a main process of deep learning, the learning method of deep neural networks, as a core technology. In addition to this, the generalized continuous form of convolution has been expressed as a new variant of Laplace-type transform that, encompasses almost all existing integral transforms. Finally, we would, in this paper, like to describe the theoretical contents as detailed as possible so that the paper may be self-contained.