Explicit construction of general multivariate Padé approximants to an Appell function

Abstract

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 (axⁱy^j/(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 $q(x,y)=(-1)^{n}{{m+n+a}\choose n}F_{1}(-m-a,-n,-n;-m-n-a;x,y)$ , where the integer m, which defines the degree of the numerator, satisfies m≥n+1 and m+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.