Explicit construction of multivariate Pade approximants to an Appell function

Properties of Padé approximants to the Gauss hypergeometric function 2 F1 (a, b; c; z) havebeen studied in several papers and some of these properties have been generalized to severalvariables in [6]. In this paper we derive explicit formulae for the general multivariate Padéapproximants to the Appell function F1 (a, 1, 1; a + 1; x, y) = ∞ =0 (ax i y j /(i + j + a)),i,jwhere a is a positive integer. In particular, we prove that the denominator of the constructedapproximant of partial degree n in x and y is given by q(x, y) = (−1)n m+n+a F1 (−m −na, −n, −n; −m−n−a; x, y), where the integer m, which defines the degree of the numerator,satisfies mn + 1 and m + a2n. This formula generalizes the univariate explicit formfor the Padé denominator of 2 F1 (a, 1; c; z), which holds for c > a > 0 and only in half ofthe Padé table. From the explicit formulae for the general multivariate Padé approximants, wecan deduce the normality of a particular multivariate Padé table.