Generalized special functions

Abstract

The special functions occurring in applied mathematics such as the binomial, exponential, logarithmic and trigonometric functions, Bessel and Whittaker functions, the error functions and the classical orthogonal polynomials are all special cases of the generalized hyper-geometric series: In chapter one, section one we start with the definition of the generalized hyper geometric function, this section also deals with the definition and properties of the G-function – the Hfunction and the H - function. In section two some of the well known integral transforms such as the Laplace transform, the Mellin transform, the Hankel transform and their inverse transforms, the H -transform introduced by Jain and Sharma, integral transforms with trigonometric kernels, finite Gegenbaur transforms are discussed. In section three some of the well known Integral equations occurring in the literature and the different methods of solution are discussed. In chapter two sixteen reduction formulae for the H -function have been established. Special cases include results given by Devra and Cook . Since H -function is the most general function of one variable, on specializing the parameters therein new and interesting results can be obtained In chapter three a number of series of H - functions are summed. Various methods are used and they are accordingly given in different sections. Some interesting special cases and recurrence relations are deduced. Some of these results generalize the results of Nair. In chapter four two Fourier series for H - function are derived. Special cases include Fourier series one each for H-function, G -function and G function. The results established here are very general in nature from which larger number of known and new results can be obtained by specializing the parameters of the H -function. Chapter five contains four differentiation formulae for the H - function. They generalize recent results established by Devra and Rathie. Special cases include known and new formulae for various special functions such as the H – function, G-function, generalized Wright hypergeometric function, generalized Wright –Bessel function etc. In chapter six a Mellin transform of H function is obtained. This result is used to obtain four Mellin transforms involving the product of two H -functions. Special cases include the results given by Gupta and Jain, Gupta and Soni and Rashmi and Jain. In chapter seven an Integral Transform involving the product of two H - functions with different arguments is evaluated. Various integral transforms namely Laplace, Mellin, Hankel, Whittaker and Meijer`s K – transforms of generalized functions can be deduced from it. Special case include a result given by Gupta. In chapter eight. four theorems on H - transforms are established. These theorems are of very general in nature. They generalize results earlier proved by Nair [6], Jain [4], Rathie, Saxena, Sharma and Varma. In chapter nine, two Integral Equations involving H - function in the kernel are solved.