Certain interesting single (or double) infinite series associated with hypergeometric functions have been expressed in terms of Psi (or Digamma) function

, for example, see Nishimoto and Srivastava [8], Srivastava and Nishimoto [13], Saxena [10], and Chen and Srivastava [5], and so on. In this sequel, with a view to unifying and extending those earlier results, we first establish two relations which some double infinite series involving hypergeometric functions are expressed in a single infinite series involving

. With the help of those series relations we derived, we next present two functional relations which some double infinite series involving

-functions, which are defined by a generalized Mellin-Barnes type of contour integral, are expressed in a single infinite series involving

. The results obtained here are of general character and only two of their special cases, among numerous ones, are pointed out to reduce to some known results.