A Study of Generalized Hypergeometric Function and its Applications in Vary Disciplines

Elementary functions, Bessel functions, Legendre functions and many other special functions are included in the large family of mathematical functions known as generalized hypergeometric functions. A power series with coefficients that are rational functions of the index defines them. They are used in many disciplines, such as engineering, statistics and physics, because of their rich mathematical features and adaptability. The beauty and interdependence of mathematical ideas are demonstrated by the Generalized Hypergeometric Function. Researchers and practitioners from a wide range of disciplines find it to be an indispensable tool due to its unifying power, rich analytical features, and broad applications. Numerous unusual functions are included as particular examples of the generalized Hypergeometric function. Legendre polynomials, Bessel functions, the confluent Hypergeometric function, and numerous more noteworthy examples are also included. An order (q+1) linear homogeneous differential equation is satisfied by the generalized hypergeometric function. In many applications, but especially in mathematical physics, this differential equation is essential. It is possible to write the generalized hypergeometric function in terms of contour integrals, which offers different representations and makes it easier to evaluate some integrals. The generalized Hypergeometric function has a wealth of transformation formulas that allow one Hypergeometric function to be transformed into another with distinct parameters. These transformations are quite useful for examining relationships between various special functions and simplifying expressions.