Explicit construction of general multivariate Padé approximants to an Appell function

Properties of Padé approximants to the Gauss hypergeometric function 2F1(a, b; c; z) have been studied in several papers and some of these properties have been generalized to several variables in [6]. In this paper we derive explicit formulae for the general multivariate Padé approximants to the Appell function F1(a, 1, 1; a + 1; x,y) = ∑ ∞ i,j=0 (axi yj /(i + j + a)), where a is a positive integer. In particular, we prove that the denominator of the constructed approximant of partial degree n in x and y is given by q(x,y) = (−1) n ( m+n+a) n F1(−m − a,−n, −n;−m−n−a; x,y), where the integer m, which defines the degree of the numerator, satisfies m � n + 1andm + a � 2n. This formula generalizes the univariate explicit form for the Padé denominator of 2F1(a, 1; c; z), which holds for c>a>0 and only in half of the Padé table. From the explicit formulae for the general multivariate Padé approximants, we can deduce the normality of a particular multivariate Padé table