E637  Nova demonstratio, quod evolutio potestatum binomii Newtoniana etiam pro exponentibus fractis valeat
(A new demonstration, with respect to which prevails the expansion of binomial powers by Newton even by fractional exponents)
Summary:
Here Euler notes the recursive relation for the general binomial coefficients, that if
(1 + x)^{a} =
S_{n = 0}^{∞}a_{
n} x^{n} and if
(a + x)^{a + 1} =
S_{n = 0}^{∞}
b_{n} x^{n},
then b_{n} = a_{n} +
a_{n  1}, generalizing the usual recursive relationship for the binomial coefficients.
Presented to the Academy on May 20, 1776.
Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 5, 1789, pp. 5258
 Opera Omnia: Series 1, Volume 16, pp. 112  121
Documents Available:
 Original publication: E637
 German translation (Artur Diener and Alexander Aycock): E637
 The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E637 include:
 Dhombres J., “Some aspects of the history of functionalequations linked to the evolution of the function concept.” Archive for History of Exact Sciences, 36 (2), pp. 91181 (1986).
 Dhombres J, Pensivy M., “Rigor and mathematical presentations in the 18thcentury  the example of proof by aepinus.” Historia Mathematica, 15 (1), pp. 931 (Feb 1988).
Return to the Euler Archive