I found a new way to calculate 4F3 by using the values of 2F1, which is easier than using the technique of differentiation under the integral sign. The easier method uses Gamma Function and summation of infinite series with the values of 2F1. And for a ≥ 2, b ≥ 2, m, n ∈ℕ and m, n are odd, by induction, We can also find that the values of 4F3 are rational linear combinations of 1, ln 2, G/π and 1/π.