In mathematics, the E-function was introduced by to extend the generalized hypergeometric series _pF_q(·) to the case p > q + 1. The underlying objective was to define a very general function that includes as particular cases the majority of the special functions known until then. However, this function had no great impact on the literature as it can always be expressed in terms of the Meijer G-function

Meijer G-Function

In mathematics, the G-function was introduced by as a very general function intended to include most of the known special functions as particular cases. This was not the only attempt of its kind: the generalized hypergeometric function and the MacRobert E-function had the same aim, but Meijer's...

, while the opposite is not true, so that the G-function is of a still more general nature.

Definition

There are several ways to define the MacRobert E-function; the following definition is in terms of the generalized hypergeometric function:

• when p ≤ q and x ≠ 0, or p = q + 1 and |x| > 1:
  
  • when p ≥ q + 2, or p = q + 1 and |x| < 1:
    
    The asterisks here remind us to ignore the contribution with index j = h as follows: In the product this amounts to replacing Γ(0) with 1, and in the argument of the hypergeometric function this amounts to shortening the vector length from p to p − 1. Evidently, this definition covers all values of p and q.
    

    Relationship with the Meijer G-function

    The MacRobert E-function can always be expressed in terms of the Meijer G-function:
    where the parameter values are unrestricted, i.e. this relation holds without exception.
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