General series identities, some additive theorems on hypergeometric functions and their applications

Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are generally applicable in nature. For the application purpose, we apply our results to some functions e.g. Trigonometric functions, Elliptic integrals, Dilogarithmic function, Error function, Incomplete gamma function, and many other special functions.