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On a special kind of algebraic functions that arise from developing the function \((1-2xz+z^2)^{1/2}\). (Ueber eine besondere Gattung algebraischer Functionen, die aus der Entwicklung der Function \((1-2xz+z^2)^{1/2}\) entstehen.) (German) ERAM 002.0059cj

Jacobi studies the polynomials \(X_n(x)\) introduced in 1784 by Legendre and proves the “fundamental property” \(X_n = 2^{-n} \cdot \frac{\partial^n(x^2-1)^n}{n! \partial x^n}\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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