Title: The scaling properties and the multiple derivative of Legendre polynomials,
Abstract: In this paper, we study the scaling properties of Legendre polynomials Pn(x).
We show that Pn(ax), where a is a constant, can be expanded as a sum of either
Legendre polynomials Pn(x) or their multiple derivatives dkPn(x)/dxk, and we
derive a general expression for the expansion coefficients. In addition, we
demonstrate that the multiple derivative dkPn(x)/dxk can also be expressed as a
sum of Legendre polynomials and we obtain a recurrence relation for the
coefficients.