Some applications of a hypergeometric identity

In this paper, we use a general identity for generalized hypergeometric series to obtain some new applications. The first application is a hypergeometric-type decomposition formula for elementary special functions and the second one is a generalization of the well-known Euler identity ei x ¼ cos x þ i sin x and an extension of hyperbolic functions in the sequel. Applying the mentioned identity on classical hypergeometric orthogonal polynomials and deriving summation formulae for some classical summation theorems are two further applications of this identity