In 1972, Andrews derived the basic analogue of Gauss's second summation theorem and Bailey's theorem by implementing basic analogue of Kummer's theorem into identity due to Jackson. Recently Lavoie et.al. derived many results closely related to Kummer's theorem, Gauss's second summation theorem and Bailey's theorem and also Rakha et. al. derive the basic analogues of results closely related Kummer's theorem. The aim of this paper is to derive basic analogues of results closely related Gauss's second summation theorem and Bailey's theorem. Some applications and limiting cases 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.
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.
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