Four Families of Summation Formulas for 4F3(1) with Application

A collection of functions organized according to their indexing based on non-negative integers is grouped by the common factor of fixed integer N. This grouping results in a summation of N series, each consisting of functions partitioned according to this modulo N rule. Notably, when N is equal to two, the functions in the series are divided into two subseries: one containing even-indexed functions and the other containing odd-indexed functions. This partitioning technique is widely utilized in the mathematical literature and finds applications in various contexts, such as in the theory of hypergeometric series. In this paper, we employ this partitioning technique to establish four distinct families of summation formulas for F34(1) hypergeometric series. Subsequently, we leverage these summation formulas to introduce eight categories of integral formulas. These integrals feature compositions of Beta function-type integrands and F23(x) hypergeometric functions. Additionally, we highlight that our primary summation formulas can be used to derive some well-known summation results.