Hypergeometric functions are one of the types of special mathematical functions, and they are used usually in the applied and engineering mathematics, especially in atomic and molecular calculations to work on the theoretical molecular structure analysis. This paper presents some new recurrence and symmetry relationships for the evaluation of hypergeometric functions. For this purpose, some mathematical applications and behaviours of hypergeometric functions are discussed, and we compare mathematical properties of different equations with help of numerical results and graphs.
Keywords: Hypergeometric functions, Gamma functions, Pochhammer symbols, Mathematica.