Pad6 Approximants for the q-Elementary Functions

There are a few particular functions whose properties under rational approximation have received special scrutiny. Exp and log are probably the central examples. This stems both from the fact that we can actually work out the details, though by no means trivially (see, for example, [9] or [11]) and from the pivotal role of these functions in applied analysis. It is also the case that almost all the known results concerning the measure of transcendence of e and ~are tied into rational approximations to exp or log [4], [8]. Our intention is to show how to construct Pad6 and related approximants to functions that satisfy particularly simple functional relations. Two examples for which this method works are the q analogues of exp and log. The q analogues of exp and log are functions parametrized by q that, in some sense, naturally reduce to exp and log on letting q tend to one. (See Sections 1 and 3.) The introduction of the q variable allows us to construct the Pad6 approximants from functional relations rather than the more usual use of the differential equations. An alternate route to some of these constructions, based on the q d algorithm, is given by Wynn [13]. Most notably, Wynn derives m>_n-1 forms of the approximants of Theorems 2 and 3. (See also [12].) The construction of q analogues of hypergeometric functions appears to be a profitable endeavor [1], [2], [5]. We might view the partial theta functions