
Our previous paper "Hypergeometric Function Representations" [15], presented an algorithm for computing formula representations of the hypergeometric function defined by
where we use notation
For example,
is a typical formula representation. Ability to compute such representations is applicable to integration, differential equations, closed form summation, and difference equations [7], [10], [13]. The Meijer G function, , defined in the next section, is a generalization of the hypergeometric function . Every hypergeometric function is a function:
However, not every function has a simple representation in terms of hypergeometric functions. In particular, Bessel functions and (), Kelvin functions and (), Whittaker function (), Lommel function (), and Legendre function () can only be represented by functions. Our new algorithm computes formula representations such as
An ability to produce such representations is crucially important to the solution of hypergeometric type integrals which appear copiously in various integral tables [5], [11], [12], [13], used by scientists and mathematicians. In this paper, we repeat some familiar themes from our previous work [15], shift operators, contiguity relations, inverse shift operators, suitable origins, accessible origins, proper sequences, and lookup certificates but in a new and different context. Just the same, the current paper is completely selfcontained and will stand on its own. 

©20042018 Planet Quantum  Kelly Roach 