The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by N{\o}rlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by B\"uhring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function

We study the group of transformations of ${}_4{F}_3$ hypergeometric functions evaluated at unity with one unit shift in parameters. We reveal the general form of this family of transformations and its group property. Next, we use explicitly known transformations to generate a subgroup whose structure is then thoroughly studied. Using some known results for ${}_3{F}_2$ transformation groups, we show that this subgroup is isomorphic to the direct product of the symmetric group of degree 5 and 5-dimensional integer lattice. We investigate the relation between two-term ${}_4{F}_3$ transformations from our group and three-term ${}_3{F}_2$ transformations and present a method for computing the coefficients of the contiguous relations for ${}_3{F}_2$ functions evaluated at unity. We further furnish a class of summation formulas associated with the elements of our group. In the appendix to this paper, we give a collection of Wolfram Mathematica^® routines facilitating the group calculations.

The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Meijer’s G function. By the application of our method to known transformation formulas, we derive about forty hypergeometric identities, the majority of which are believed to be new.

In this paper we continue investigation of the hypergeometric function \({}_{4}F_{3}(1)\) as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters exceeds one of the bottom parameters by a positive integer or, reversely, one of the bottom parameters exceeds one of the top parameters by a positive integer, or both. We show that all such cases reduce to the cases with the unit parameter differences. The latter case, in turn, can be expressed in terms of certain linear combination of two parameter derivatives of the function \({}_{3}F_{2}(1)\) or, equivalently, of two series involving the logarithmic derivative of the gamma function.