[Paper Review] The scaling properties and the multiple derivative of Legendre polynomials

Guillaume Laurent, Geoffrey Robert Harrison|arXiv (Cornell University)|Oct 28, 2017

Fractional Differential Equations Solutions3 references3 citations

TL;DR

This paper derives explicit analytical expressions for the scaling of Legendre polynomials $P_n(\lambda x)$ and their multiple derivatives. It shows that $P_n(\lambda x)$ can be expanded as a sum of either Legendre polynomials or their derivatives, with closed-form coefficients derived via Rodrigues' formula and hypergeometric series, and establishes a recurrence relation for the coefficients of the derivative expansion.

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.

Motivation & Objective

• To derive explicit expressions for the scaling transformation $P_n(\lambda x)$, where $\lambda$ is a constant, extending known properties of Legendre polynomials.
• To express $P_n(\lambda x)$ as a linear combination of Legendre polynomials $P_{n-2k}(x)$ and their multiple derivatives $d^kP_{n-k}/dx^k$, enabling analytical treatment in physical and mathematical problems.
• To establish a recurrence relation for the coefficients in the expansion of the $k$-th derivative of $P_n(x)$ in terms of lower-degree Legendre polynomials.
• To provide a systematic method for computing expansion coefficients using Rodrigues' formula, binomial expansion, and hypergeometric series identities.
• To support applications in physics and engineering, particularly in the projection of spherical harmonics onto planes, by offering analytical tools for scaled Legendre functions.

Proposed method

• Derives $P_n(\lambda x)$ using Rodrigues' formula and applies the chain rule and binomial expansion to express the scaled polynomial as a sum over derivatives of lower-degree Legendre polynomials.
• Expands the resulting polynomial in $x^2 - 1$ terms using Vieta’s formula to determine coefficients involving $\lambda$ and $n$.
• Applies the $n$-th derivative to the expanded polynomial and rewrites the result in terms of Legendre polynomials via their Rodrigues representation.
• Derives the expansion of the $k$-th derivative $d^kP_n/dx^k$ as a sum of Legendre polynomials by matching coefficients in hypergeometric series representations.
• Uses the hypergeometric function representation $P_n(x) = {}_{2}F_1(-n,n+1,1;(1-x)/2)$ and applies differentiation rules to obtain a series for $d^kP_n/dx^k$.
• Establishes a recurrence relation for the expansion coefficients $\alpha_{n-k-2i}$ by equating coefficients of like powers in the hypergeometric series expansions.

Experimental results

Research questions

• RQ1 How can $P_n(\lambda x)$ be expressed as a linear combination of Legendre polynomials $P_{n-2k}(x)$ for a constant scaling factor $\lambda$?
• RQ2 What is the analytical form of the expansion coefficients $b_{\lambda,n,k}$ in the representation $P_n(\lambda x) = \sum_k b_{\lambda,n,k} P_{n-2k}(x)$?
• RQ3 Can the $k$-th derivative of a Legendre polynomial $d^kP_n/dx^k$ be expressed as a finite sum of lower-degree Legendre polynomials, and if so, what are the coefficients?
• RQ4 What recurrence relation governs the coefficients $\alpha_{n-k-2i}$ in the expansion $d^kP_n/dx^k = \sum_i \alpha_{n-k-2i} P_{n-k-2i}(x)$?
• RQ5 How do the scaling and derivative properties of Legendre polynomials relate to their hypergeometric representation and falling factorial identities?

Key findings

• The scaling property $P_n(\lambda x)$ is expressed as $\sum_{k=0}^{\lfloor n/2 \rfloor} a_{\lambda,n,k} \frac{d^k}{dx^k} P_{n-k}(x)$, with coefficients $a_{\lambda,n,k} = \frac{\lambda^{n-2k}(\lambda^2 - 1)^k}{2^k k!}$.
• The expansion of $P_n(\lambda x)$ in terms of Legendre polynomials is given by $P_n(\lambda x) = \sum_{k=0}^{\lfloor n/2 \rfloor} b_{\lambda,n,k} P_{n-2k}(x)$, where $b_{\lambda,n,k}$ is derived from a double sum involving the coefficients $\alpha_{n,k,i}$.
• The $k$-th derivative of $P_n(x)$ is shown to be expressible as a finite sum: $\frac{d^k}{dx^k} P_n(x) = \sum_{i=0}^{\lfloor (n-k)/2 \rfloor} \alpha_{n-k-2i} P_{n-k-2i}(x)$, with coefficients $\alpha_{n-k-2i}$ satisfying a recurrence relation.
• The recurrence for $\alpha_{n-k-2i}$ is derived as $\alpha_{n-k-2i} = \frac{2^{k+2i}(n-\frac{1}{2})^{\underline{k}}(n-i)^{\underline{i}}(n-k-\frac{1}{2})^{\underline{2i}}}{(2i)^{\underline{2i}}(n-\frac{1}{2})^{\underline{i}}} - \sum_{l=0}^{i-1} \frac{(2(n-k-i-l))^{\underline{2(i-l)}}}{(2(i-l))^{\underline{2(i-l)}}} \alpha_{n-k-2l}$.
• The coefficients $\alpha_{n,k,i}$ are explicitly given by $\alpha_{n,k,i} = \frac{2^{k+i}(n-k+i-\frac{1}{2})^{\underline{k-i}}(n-k)^{\underline{i}}(n-2k+2i-\frac{1}{2})^{\underline{2i}}}{(2i)^{\underline{2i}}(n-k+i-\frac{1}{2})^{\underline{i}}} - \sum_{l=0}^{i-1} \frac{(2(n-2k+i-l))^{\underline{2(i-l)}}}{(2(i-l))^{\underline{2(i-l)}}} \alpha_{n,k-i+l,l}$.
• The derivation confirms that the $k$-th derivative of $P_n(x)$ is a finite linear combination of Legendre polynomials of degree $n-k-2i$, with coefficients computable via recurrence, enabling exact analytical treatment in physical applications.