On the relationship between the generalized hypergeometric function and Meijer's G-function

At present, we know that the generalized hypergeometric function is a Stieltjes integral with Meijer's G function density. On the other hand,  Meijer's G function is the sum of the products of hypergeometric and power functions. We will discuss the consequences of such representations. Particular attention will be paid to the analytic continuation of the   G -function from the interior to the exterior of the unit disk. Simple formulas for such a continuation will be presented. We also study the behavior of the principal on the banks of the branch cut.

Similarly to the Miller-Paris transformations for the generalized hypergeometric functions, we present identities for the Meijer G-function having integer parameter differences. Next, we give some curious integral involving G-function and its corollaries in the form of summation formulas. The report is based on the results of joint work with Dmitrii Karp.