Closed-form formulas of two Gauss hypergeometric functions of specific parameters

Abstract

Using the Faà di Bruno formula, along with three identities of the partial Bell polynomials, and leveraging two differentiation formulas for the Gauss hypergeometric functions, the authors present several closed-form formulas for the Gauss hypergeometric functions${}_{2}F_1(n+\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};-z^2)\text{and}{}_{2}F_1(1,n+\frac{1}{2};n+\frac{3}{2};z^2)$ for $n\in \{0,1,2,\dots \}$ and $\left|z\right|<1$. These formulas are analyzed in light of three Gauss relations for contiguous functions, with the aid of a relation between the Gauss hypergeometric functions and the Lerch transcendent. Additionally, the authors determine the location and distribution of the zeros of two polynomials involved in these representations, which contain generalized binomial coefficients. By comparing these formulas, they also derive several combinatorial identities.