Brief history
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A rational function matching a formal power series, at least to the term equal to the sum of the denominator and numerator degree, is called a Padé approximant of that series. The name was given after Henri Padé (1863 - 1953), who did the first systematic study of this rational approximation in his Ph.D. thesis (1892) [1]. However, in the XVIII century, this technique was discovered, independently, by Johann Lambert (1728 - 1777) and Joseph Lagrange (1736 - 1813). Furthermore, Carl Jacobi (1804 - 1851) presented a ratio of determinants to calculate a Padé approximant, and George Frobenius (1849 - 1917) proved recursive relations between adjacent approximants. 

After Padé's thesis, in the 1960s, the Padé approximants come back to the scene with Baker's works in mathematical physics [2]. 

By the end of the XX century, over 6000 references were estimated about its application in several areas, such as, number theory, differential equations, numerical analysis, and fluid mechanics. For a more detailed history see [3].

Claude Brezinski [4] is one of the most influential of the XX and XXI centuries in the study of Padé approximants.




.. [1] Padé, H. (1892). Sur la représentation approchée d’une fonction par des fractions rationnelles. Annales scientifiques de l’École normale supérieure, 9(9):3–93.

.. [2] Baker, G. A. J. (1975). Essentials of Padé Approximants. Academic Press.

.. [3] Brezinski, C. (1991). History of Continued Fractions and Padé Approximants. Springer Series in Computational Mathematics, Verlag Berlin Heidelberg.

.. [4] https://math.univ-lille1.fr/~brezinsk/