The hypergeometric functions are one of the most important and special functions in mathematics. They are the generalization of the exponential functions. Particularly the ordinary hypergeometric function 2F1(a, b; c; z) is represented by hypergeometric series and is a solution to a second order differential equation. Similarly, Laplace transform is a form of integral transform that converts linear differential equations to algebraic equations. This paper aims to study the convergence of hypergeometric function and Laplace transform of some hypergeometric functions. Moreover, some relationships between Laplace transformation and hypergeometric functions is established in the concluding section of this paper.

The study of flow of liquids in a circular pipe has been studied for a long time. The theoretical and numerical analysis of viscous fluid is very important in the field of physics, engineering and even in medicine. In any fluid flow, the Napier-Stokes equation are very important to study the nature of flow. On the basis of Reynold number, the flow is either classified as laminar or turbulent. When the heat is supplied to a circular pipe with a liquid having laminar flow, the velocity, rate of flow of volume, temperature gradient, etc. are changed. This study aims to investigate the change in velocity, pressure drop, frictional factor and temperature distribution in the thermal layer across the liquid in the laminar flow. Various boundary conditions are assumed and the conservation of energy, momentum are also considered.

The eastern society is rich in terms of science and technology. Mathematics is considered as the base of science. The eastern history shows that the Hindu society is rich in mathematics. The evolution of mathematics can be studied from the time of 'patiganita' to the latest form of science and technology. Geometric series is an important tool in arithmetic which is now developed to hypergeometric function. Hypergeometric function is an advance function used to solve differential equations of second order. The purpose of this paper is to find the linkage between the ancient geometric series (Gunanka sreni) to the modern Hypergeometric function and to expose the work of ancient Hindu mathematicians which is believed to be narrow among the mathematics researchers of the present period. In this paper, some forms of geometric series that were used in Hindu mathematics are interpreted in terms of hypergeometric series.