General order multivariate Padé approximants for pseudo-multivariate functions

Although general order multivariate Padé approximants have been introduced some decades ago, very few explicit formulas have been given so far. We show in this paper that, for any given pseudo-multivariate function, we can compute its general order multivariate Padé approximant for some given index sets with the usage of Maple or other software. Examples include a multivariate form of the sine function ! #" $&%! '( *) ) +, -* /. * %! 0 a multivariate form of the logarithm function 1 #  '( 32 4 -5 6. a multivariate form of the inverse tangent function 7 8 # 9 8 : ;" <$&%! '( *) ) += -* 6.> ( %

[1]  Annie Cuyt,et al.  How well can the concept of Padé approximant be generalized to the multivariate case , 1999 .

[2]  Bernhard Beckermann,et al.  SOME EXPLICIT FORMULAS FOR PADS APPROXIMANTS OF RATIOS OF HYPERGEOMETRIC FUNCTIONS , 2008 .

[3]  B. Dwork.  Generalized Hypergeometric Functions , 1990 .

[4]  Kathy Driver,et al.  Kronecker type theorems, normality and continuity of the multivariate Padé operator , 1996 .

[5]  Edward B. Saff,et al.  Convergence of Padé approximants of partial theta functions and the Rogers-Szegö polynomials , 1987 .

[6]  Annie A. M. Cuyt,et al.  Exploring multivariate Padé approximants for multiple hypergeometric series , 1999, Adv. Comput. Math..

[7]  Henri Padé.  Recherches sur la convergence des développements en fractions continues d'une certaine catégorie de fonctions , 1907 .

[8]  Ping Zhou.  Explicit construction of multivariate Pade´ approximants , 1997 .

[9]  D. Levin,et al.  General Order Padè-type Rational Approximants Defined from Double Power Series , 1976 .

[10]  Peter Borwein,et al.  Pad6 Approximants for the q-Elementary Functions , 1988 .

[11]  Ping Zhou.  Multivariate Padé Approximants Associated with Functional Relations , 1998 .

[12]  Kathy Driver,et al.  A direct approach to convergence of multivariate, nonhomogeneous, Pade´ approximants , 1996 .

[13]  Peter B. Borvein.  Padé approximants for theq-elementary functions , 1988 .

[14]  Hassane Allouche,et al.  On the structure of a table of multivariate rational interpolants , 1992 .

[15]  Ping Zhou.  Explicit Construction of Multivariate Padé Approximants for a q-Logarithm Function , 2000 .

[16]  Annie A. M. Cuyt,et al.  Explicit construction of general multivariate Padé approximants to an Appell function , 2005, Adv. Comput. Math..