General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the $q$-exponential functions

$E\left( x,y\right) =\sum_{i,j=0}^\infty \frac{x^iy^j}{\left( i+j\right) !}$

and

$E_q\left( x,y\right) =\sum_{i,j=0}^\infty \frac{x^iy^j}{[i+j]_q!},$

as well as the Appell function

$F_1\left( a,1,1;c;x,y\right) =\sum_{i,j=0}^\infty \frac{\left( a\right) _{i+j}x^iy^j}{\left( c\right) _{i+j}}$

and the multivariate form of the partial theta function

$F\left( x,y\right) =\sum_{i,j=0}^\infty q^{-\left( i+j\right) ^2/2}x^iy^j.$