Meijer G–Functions: A Gentle Introduction

T he Meijer G–functions are a remarkable family G of functions of one variable, each of them determined by finitely many indices. Although each such function is a linear combination of certain special functions of standard type, they seem not to be well known in the mathematical community generally. Indeed they are not even mentioned in most books on special functions, e.g., [1], [18]. Even the new comprehensive treatise [15] devotes a scant 2 of its 900+ pages to them. (The situation is different in some of the literature oriented more toward applications, e.g., the extensive coverage in [6] and [9].) The present authors were ignorant of all but the name of the G–functions until the second author found them relevant to his research [3]. As we became acquainted with them, we became convinced that they deserved a wider audience. Some reasons for this conviction are the following: • The G–functions play a crucial role in a certain useful mathematical enterprise. • When looked at conceptually, they are both natural and attractive. • Most special functions, and many products of special functions, are G–functions or are expressible as products of G–functions with elementary functions. There are seventy-five such formulas in [4, sec. 5.6]; see also [6, sec. 6.2], [9, chap. 2], and [20]. Examples are the exponential function, Bessel functions, and products of Bessel functions (the notation will be explained below):