Paul Émile Appell

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Paul Appell (September 27, 1855 in Strasbourg – October 24, 1930 in Paris), also known as Paul Émile Appel, was a French mathematician and Rector of the University of Paris. The concept of Appell polynomials is named after him, as is rue Paul Appell in the 14th Arrondissement in Paris.

Paul Appell entered the École Normale Supérieure in 1873. He was elected to the French Academy of Sciences in 1892.

Between 1903 and 1920 he was Dean of the Faculty of Science of the University of Paris, then Rector of the University of Paris from 1920 to 1925.

He worked first on projective geometry in the line of Chasles, then on algebraic functions, differential equations, and complex analysis.

He has introduced a set of four hypergeometric series F1,F2,F3,F4 of two variables that generalize Gauss's hypergeometric series.

The Appell series F1 is defined for | x | , | y | < 1 by the double series:

F_1(a,b,c,d,x,y)=\sum_{n,m=0}^\infty \frac{(a,m+n)(b,m)(c,n)}{(d,m+n)(1,m)(1,n )} x^m y^n,

where (q,n) is the Pochammer symbol (q,0) = 1 and (q,n)=q(q+1)\ldots(q+n-1)=\frac{\Gamma(q+n)}{\Gamma(q)} for n > 0. For other values of x,y the function F1 can be defined by analytic continuation.

The function F2 is defined by the series:

F_2(a,b_1,b_2,c_1,c_2,x,y)=\sum_{n,m=0}^\infty \frac{(a,m+n)(b_1,m)(b_2,n)}{(c _1,m)(c_2,n)(1,m)(1,n)} x^m y^n,

the function F3, by the series:

F_3(a_1,a_2,b_1,b_2,c,x,y)=\sum_{n,m=0}^\infty \frac{(a_1,m)(a_2,n)(b_1,m)(b_2 ,n)}{(c,m+n)(1,m)(1,n)} x^m y^n,

and the function F4, by the series:

F_4(a,b,c_1,c_2,x,y)=\sum_{n,m=0}^\infty \frac{(b,m+n)(a,m+n)}{(c_1,m)(c_2,n)( 1,m)(1,n)} x^m y^n,

He established the set of partial differential equations of which these functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. In 1926, with Joseph-Marie Kampé de Fériet, he authored a treatise on generalized hypergeometric series.

In mechanics, he proposed an alternative formulation of analytical mechanics known as Appell's equation of motion.

He discovered a physical interpretation of the imaginary period of the doubly periodic function whose restriction to real arguments describes the motion of an ideal pendulum.

His daughter Marguerite Appell (1883–1969), who married the mathematician Émile Borel, is known as a novelist under her pen-name Camille Marbo.

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  • (fr:) P. Appell, "Notice sur les travaux scientifiques" Acta Mathematica 45 (1925) pp. 161–285. describes 257 of Appell's publications.
  • (fr:) E. Lebon, Biographie et bibliographie analytique des écrits de Paul Appell (Paris, 1910)
  • (fr:) P. Appell, "Sur une classe de polynômes", Annales scientifiques de l'École Normale Supérieure 2e série, tome 9, 1880.
  • (fr:) P. Appell, "Sur les fonctions hypergeometriques de deux variables" Journal de Mathématiques Pures et Appliquées series III,8, 173 (1882).
  • (fr:) P. Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, volume 87, number 1, July, 1878.

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