  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.
