General order multivariate Padé approximants for Pseudo-multivariate functions. II

Abstract

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E ( x _ ) = ∑ j 1 , j 2 , … , j m = 0 ∞ x 1 j 1 x 2 j 2 ⋯ x m j m ( j 1 + j 2 + ⋯ + j m ) ! , \begin{equation*} E\left (\underline {x}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty } \frac {x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{\left ( j_{1}+j_{2}+\cdots +j_{m}\right ) !}, \end{equation*} the logarithm function L ( x _ ) = ∑ j 1 + j 2 + ⋯ + j m ≥ 1 x 1 j 1 x 2 j 2 ⋯ x m j m j 1 + j 2 + ⋯ + j m , \begin{equation*} L(\underline {x})=\sum _{j_{1}+j_{2}+\cdots +j_{m}\geq 1}\frac { x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{j_{1}+j_{2}+\cdots +j_{m}}, \end{equation*} the Lauricella function F D ( m ) ( a , 1 , … , 1 ; c ; x 1 , … , x m ) = ∑ j 1 , j 2 , … , j m = 0 ∞ ( a ) j 1 + ⋯ + j m ( c ) j 1 + ⋯ + j m x 1 j 1 ⋯ x m j m , \begin{equation*} F_{D}^{\left ( m\right ) }\left ( a,1,\ldots ,1;c;x_{1},\ldots ,x_{m}\right ) =\sum _{j_{1},j_{2},\ldots ,j_{m}=0}^{\infty }\frac {\left ( a\right ) _{j_{1}+\cdots +j_{m}}}{\left ( c\right ) _{j_{1}+\cdots +j_{m}}} x_{1}^{j_{1}}\cdots x_{m}^{j_{m}}, \end{equation*} and many more. We prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives. These properties do not hold in general for multivariate Padé approximants. A truncation error upperbound is also given.