An update on the Upadhyaya transform

Abstract

It is almost two years now when the Upadhyaya transform was introduced  by the first author (Upadhyaya, Lalit Mohan, Introducing the Upadhyaya integral transform,  Bull. Pure Appl. Sci. Sect. E Math. Stat., 38(E)(1), 471–510, doi 10.5958/2320-3226.2019.00051.1 https://www.researchgate.net/publication/334033797)) as the  most powerful, versatile and robust generalization and unification of a number of variants  of the classical Laplace transform which have appeared in the mathematics research literature  during the years 1993 to 2019. In this paper we provide an update on the Upadhyaya  transform, where we explain the definition the one-dimensional Upadhyaya transform and  its n-dimensional generalization in more detail and we show that how many other various  variants of the classical Laplace transform, that have come to our notice since then and  most of which are introduced into the mathematics research literature during the past  two years by a number of authors after the advent of the Upadhyaya transform, follow  as a special case of the Upadhyaya transform. We find the Upadhyaya transform of some  trigonometric and hyperbolic functions, the sine integral, the generalized hypergeometric  function and the Bessel function of the first kind in order to exemplify the vast power of  the Upadhyaya transform and we also correct a minor typo in the aforementioned paper  of the first author.