Meijer G –Functions: A Gentle Introduction Richard Beals and Jacek Szmigielski 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 func- tion 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 express- ible 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): Richard Beals is emeritus professor of mathematics at Yale University. His email address is richard.beals@yale.edu. Jacek Szmigielski is professor of mathematics at the Univer- sity of Saskatchewan. His email address is szmigiel@math. usask.ca. DOI: http://dx.doi.org/10.1090/notimanid1016 e x = G 10 01  1      − x  , J ν (2x) = x −ν G 10 02  ν 0      x 2  , J 2μ (x) J 2ν (x) = 1 √ π G 12 24  1 2 0 μ + ν ν − μ μ − ν −μ − ν      x 2  . • The family G of G–functions has remarkable closure properties: it is closed under the reflections x →−x and x → 1/x, multiplication by powers, differentiation, integration, the Laplace transform, the Euler transform, and the multiplicative con- volution. Thus, if G, G 1 , and G 2 belong to G and the various transforms and the multiplicative convolution G 1 ∗ G 2 exist, then the following also belong to G: G(−x), G(1/x), x a G(x), G ′ (x); (1)  x c G(y)dy (for some choice of c ); (2) LG(x) ≡  ∞ 0 e −xy G(y)dy ; (3) E a,b G(x) ≡  1 0 t a−1 (1 − t) b−1 G(ty)dt ; (4) [G 1 ∗ G 2 ](x) ≡  ∞ 0 G 1  x y  G 2 (y) dy y . (5) • The family G is minimal with respect to these properties. For example, the only nonzero multiple of e x that belongs to G is e x itself. Closure under convolution, (5), is of particu- lar importance for the mathematical enterprise alluded to above. It lies at the heart of the most comprehensive tables of integrals in print [16] and online, as well as the Mathematica integrator; see [19], [6], and [8]. 866 Notices of the AMS Volume 60, Number 7