What are Legendre polynomials and after whom are they named?

Legendre polynomials are a system of complete and orthogonal polynomials in mathematics, named after Adrien-Marie Legendre, who introduced them in 1782. They possess a wide array of mathematical properties and are utilized in numerous applications across various fields.

What are some mathematical concepts closely related to Legendre polynomials?

Closely related mathematical concepts include associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. These are often generalizations or specific variations that share fundamental properties with Legendre polynomials.

How are the first six Legendre polynomials visually represented?

The source material includes a visual representation, specifically a plot, illustrating the graphs of the first six Legendre polynomials. These plots show the characteristic curves of the polynomials for different degrees.

How are Legendre polynomials defined through their construction as an orthogonal system?

In this approach, Legendre polynomials are defined as an orthogonal system with respect to a weight function of w(x)=1 over the interval [-1,1]. This means that the integral of the product of two different Legendre polynomials over this interval is zero.

What is the orthogonality condition for Legendre polynomials P_n(x) and P_m(x) when defined as an orthogonal system?

The orthogonality condition states that the integral from -1 to 1 of P_m(x)P_n(x) dx equals 0 if n is not equal to m. This property is fundamental to their use in representing functions.

What standardization condition is applied to uniquely determine Legendre polynomials?

Legendre polynomials are uniquely determined by the additional standardization condition that P_n(1)=1. This condition fixes the scaling of each polynomial.

How are the first few Legendre polynomials, P_0(x) and P_1(x), constructed based on orthogonality and standardization?

P_0(x) is constructed as 1, being the only correctly standardized polynomial of degree 0. P_1(x) is determined by requiring orthogonality to P_0, which leads to P_1(x)=x. Higher-degree polynomials are then built by demanding orthogonality to all preceding polynomials and applying the standardization condition.

What are the advantages of defining Legendre polynomials via their construction as an orthogonal system?

This definition is considered the simplest, as it does not rely on the theory of differential equations. It immediately demonstrates the completeness of the polynomials, and it establishes them as one of the three classical orthogonal polynomial systems.

How do Legendre polynomials compare to Laguerre and Hermite polynomials as classical orthogonal polynomial systems?

Legendre polynomials are orthogonal over the interval [-1,1] with a weight function of 1. Laguerre polynomials are orthogonal over the half-line [0,infinity) with a weight of e^(-x), while Hermite polynomials are orthogonal over the full line (-infinity, infinity) with a weight of e^(-x^2).

How can Legendre polynomials be defined using a generating function?

Legendre polynomials can be defined as the coefficients in a formal expansion in powers of 't' of the generating function 1/sqrt(1-2xt+t^2) = sum from n=0 to infinity of P_n(x)t^n. This method provides a compact way to express the entire sequence of polynomials.

What are the first two Legendre polynomials derived from the generating function expansion?

Expanding the generating function up to t^1 yields P_0(x)=1 and P_1(x)=x. These initial polynomials serve as the basis for generating all subsequent polynomials.

What is Bonnet's recursion formula and how is it derived from the generating function?

Bonnet's recursion formula is (n+1)P_n+1(x) = (2n+1)xP_n(x) - nP_n-1(x). It is obtained by differentiating the generating function with respect to 't', rearranging the terms, and equating the coefficients of powers of 't'. This formula allows for the recursive generation of all Legendre polynomials given the first two.

What is the historical significance of the generating function approach for Legendre polynomials?

The generating function approach is directly linked to the multipole expansion in electrostatics and is the method by which Adrien-Marie Legendre first defined these polynomials in 1782. This highlights their origin in physical applications.

How are Legendre polynomials defined in terms of a differential equation?

Legendre polynomials are defined as solutions to Legendre's differential equation: (1-x^2)P_n''(x) - 2xP_n'(x) + n(n+1)P_n(x) = 0. This equation is a second-order linear ordinary differential equation.

What are the characteristics of the solutions to Legendre's differential equation?

Legendre's differential equation has regular singular points at x = +/-1. When 'n' is an integer, the solution P_n(x) that is regular at x=1 is also regular at x=-1, and this series terminates, resulting in a polynomial. There is also a non-polynomial solution known as Legendre functions of the second kind, Q_n.

How does Sturm-Liouville theory relate to Legendre's differential equation and its solutions?

Legendre's differential equation can be rewritten as an eigenvalue problem, making it a Sturm-Liouville equation. From this framework, the orthogonality and completeness of the Legendre polynomial solutions (eigenfunctions) are evident, with eigenvalues of the form n(n+1).

What is Rodrigues' formula for Legendre polynomials?

Rodrigues' formula provides a compact expression for Legendre polynomials: P_n(x) = (1 / (2^n * n!)) * (d^n / dx^n) * (x^2 - 1)^n. This formula is powerful for deriving many properties of these polynomials.

What are the first few Legendre polynomials, from P_0(x) to P_5(x)?

The first few Legendre polynomials are: P_0(x) = 1, P_1(x) = x, P_2(x) = (1/2)(3x^2 - 1), P_3(x) = (1/2)(5x^3 - 3x), P_4(x) = (1/8)(35x^4 - 30x^2 + 3), and P_5(x) = (1/8)(63x^5 - 70x^3 + 15x).

What is the combined statement of orthogonality and normalization for Legendre polynomials?

The orthogonality and normalization of Legendre polynomials can be compactly expressed as the integral from -1 to 1 of P_m(x)P_n(x) dx = (2 / (2n+1)) * delta_mn, where delta_mn is the Kronecker delta, which is 1 if m=n and 0 otherwise.

What does the completeness property of Legendre polynomials imply for piecewise continuous functions?

The completeness property means that any piecewise continuous function f(x) with finitely many discontinuities in the interval [-1,1] can be approximated in the mean by a sequence of sums of Legendre polynomials. The coefficients for this expansion are given by a_l = ((2l+1)/2) * integral from -1 to 1 of f(x)P_l(x) dx.

How were Legendre polynomials initially applied in physics?

Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre as the coefficients in the expansion of the Newtonian potential, which describes the gravitational or Coulomb potential due to a point mass or charge. This expansion is useful for integrating over continuous mass or charge distributions.

How do Legendre polynomials appear in the solution of Laplace's equation in spherical coordinates?

When solving Laplace's equation for static potential in a charge-free region using separation of variables in spherical coordinates, especially with axial symmetry, the solution for the potential Phi(r,theta) is expressed as a sum involving Legendre polynomials: Phi(r,theta) = sum from l=0 to infinity of (A_l*r^l + B_l*r^-(l+1))P_l(cos theta).

What is the role of Legendre polynomials in multipole expansions?

Legendre polynomials are crucial in multipole expansions, where they are used to expand functions of the form 1/sqrt(1+eta^2-2eta*x). This form naturally arises in calculating electric potentials, for instance, from a point charge. The left-hand side of this equation is the generating function for Legendre polynomials.

What happens in the multipole expansion if the observation point is closer to the charge than its location?

If the radius 'r' of the observation point is smaller than 'a' (the charge's location), the potential can still be expanded using Legendre polynomials, but with 'a' and 'r' exchanged in the expansion terms. This is the foundation for the interior multipole expansion.

How are trigonometric functions like cos(n*theta) related to Legendre polynomials?

Trigonometric functions such as cos(n*theta), which are also represented by Chebyshev polynomials T_n(cos theta), can be expanded using Legendre polynomials P_n(cos theta). For example, T_0(cos theta) = P_0(cos theta) and T_1(cos theta) = P_1(cos theta).

How are Legendre polynomials utilized in recurrent neural networks?

In recurrent neural networks, Legendre polynomials are used to optimize networks containing a d-dimensional memory vector. The neural activities can be made to obey a linear time-invariant system, where the sliding window of the input is approximated by a linear combination of shifted Legendre polynomials. This approach can lead to improved performance with fewer computational resources compared to other architectures like long short-term memory units.

What is the parity property of Legendre polynomials?

Legendre polynomials have definite parity, meaning they are either even or odd functions. This property is expressed as P_n(-x) = (-1)^n P_n(x), indicating that the polynomial is even if 'n' is even and odd if 'n' is odd.

What is the integral of a Legendre polynomial over the interval [-1,1] for n >= 1?

For any Legendre polynomial P_n(x) where n is greater than or equal to 1, its integral over the interval [-1,1] is 0. This property is a direct consequence of their orthogonality with P_0(x)=1.

How is the average of a function approximated by a Legendre series related to its coefficients?

When a function or experimental data is approximated by a Legendre series, the average of that series over the interval [-1,1] is simply given by the leading expansion coefficient, a_0. This simplifies the calculation of the average value.

What is the underivative formula for Legendre polynomials?

The underivative of a Legendre polynomial P_n(x) for n >= 1 is given by the formula: integral P_n(x) dx = (1 / (2n+1)) * [P_n+1(x) - P_n-1(x)]. This formula relates the integral of a polynomial to other polynomials in the sequence.

What is the value of a Legendre polynomial P_n(x) at x=1 and x=-1?

Due to their parity and standardization, Legendre polynomials have specific values at the boundaries: P_n(1) = 1 and P_n(-1) = (-1)^n. This means P_n(-1) is 1 for even 'n' and -1 for odd 'n'.

What are the values of Legendre polynomials at the origin (x=0)?

At the origin, the values are P_2n(0) = ((-1)^n / 4^n) * (2n choose n) and P_2n+1(0) = 0. This shows that odd-degree Legendre polynomials are zero at the origin, while even-degree ones have non-zero values.

What is the derivative of a Legendre polynomial at the endpoint x=1?

The derivative of a Legendre polynomial P_n(x) at the endpoint x=1 is given by P_n'(1) = n(n+1)/2. This provides a specific value for the slope of the polynomial at that point.

What is the Askey-Gasper inequality for Legendre polynomials?

The Askey-Gasper inequality states that the sum of Legendre polynomials from j=0 to n of P_j(x) is greater than or equal to 0 for x >= -1. This inequality provides a lower bound for the sum of these polynomials.

How can Legendre polynomials of a scalar product of unit vectors be expanded using spherical harmonics?

The Legendre polynomials of a scalar product of unit vectors (r * r') can be expanded using spherical harmonics through the formula: P_l(r * r') = (4pi / (2l+1)) * sum from m=-l to l of Y_lm(theta,phi) * Y_lm*(theta',phi'). Here, Y_lm represents the spherical harmonics, and (theta,phi) and (theta',phi') are the spherical coordinates of the unit vectors.

What are the properties of the zeros of Legendre polynomials?

All 'n' zeros of a Legendre polynomial P_n(x) are real, distinct from each other, and lie within the open interval (-1,1). Additionally, if x_k is a zero, then -x_k is also a zero, reflecting their parity property.

What is the interlacing property of Legendre polynomial zeros?

The interlacing property states that if the interval [-1,1] is divided into n+1 subintervals by the zeros of P_n(x), then each of these subintervals will contain exactly one zero of P_n+1(x). This describes how the zeros of successive polynomials are distributed.

What is the significance of Legendre polynomial zeros in numerical integration?

The zeros of Legendre polynomials play a crucial role in numerical integration, specifically in a method known as Gaussian quadrature. The particular quadrature method based on these polynomials is called Gauss-Legendre quadrature, which provides highly accurate approximations of integrals.

What are shifted Legendre polynomials and how are they defined?

Shifted Legendre polynomials, denoted as P_tilde_n(x), are defined as P_n(2x-1). This definition uses an affine transformation that bijectively maps the interval [0,1] to the interval [-1,1], effectively 'shifting' the domain of the original Legendre polynomials.

What is the orthogonality property of shifted Legendre polynomials?

The shifted Legendre polynomials P_tilde_n(x) are orthogonal over the interval [0,1]. Their orthogonality integral is given by the integral from 0 to 1 of P_tilde_m(x)P_tilde_n(x) dx = (1 / (2n+1)) * delta_mn.

What is Rodrigues' formula for the shifted Legendre polynomials?

The analogue of Rodrigues' formula for the shifted Legendre polynomials is P_tilde_n(x) = (1 / n!) * (d^n / dx^n) * (x^2 - x)^n. This formula provides a direct way to compute these polynomials.

What are the first few shifted Legendre polynomials, from P_tilde_0(x) to P_tilde_5(x)?

The first few shifted Legendre polynomials are: P_tilde_0(x) = 1, P_tilde_1(x) = 2x-1, P_tilde_2(x) = 6x^2-6x+1, P_tilde_3(x) = 20x^3-30x^2+12x-1, P_tilde_4(x) = 70x^4-140x^3+90x^2-20x+1, and P_tilde_5(x) = 252x^5-630x^4+560x^3-210x^2+30x-1.

What are Legendre rational functions and how are they constructed?

Legendre rational functions are a sequence of orthogonal functions defined on the interval [0, infinity). They are constructed by composing the Cayley transform with the standard Legendre polynomials, specifically R_n(x) = (sqrt(2) / (x+1)) * P_n((x-1)/(x+1)).

What kind of differential problem are Legendre rational functions eigenfunctions of?

Legendre rational functions are eigenfunctions of a singular Sturm-Liouville problem, which is expressed as (x+1) * d/dx * (x * d/dx * [(x+1)v(x)]) + lambda*v(x) = 0. The eigenvalues for this problem are lambda_n = n(n+1).

What is Hilb's formula, and what does it describe for Legendre polynomials?

Hilb's formula describes the asymptotic behavior of Legendre polynomials P_l(cos theta) as l approaches infinity. It provides an approximation involving Bessel functions J_0 and J_1, showing how the polynomials behave for large degrees.

What is the general recurrence relation for the coefficients of a Legendre polynomial when expressed as a power series?

When a Legendre polynomial P_n(x) is expressed as a power series sum a_n,k * x^k, the coefficients a_n,k can be calculated using the recurrence relation a_n,k = -((n-k+2)(n+k-1) / (k(k-1))) * a_n,k-2. This allows for systematic determination of the coefficients.

What is the relationship between the derivative of P_n+1(x) and other Legendre polynomials?

The derivative of P_n+1(x) can be expressed as a sum of lower-degree Legendre polynomials: d/dx P_n+1(x) = (2n+1)P_n(x) + (2(n-2)+1)P_n-2(x) + (2(n-4)+1)P_n-4(x) + ... This shows how derivatives can be decomposed into the orthogonal basis.

What is the recurrence relation that connects the derivative of P_n(x) to P_n(x) and P_n-1(x)?

A recurrence relation for the derivative is ((x^2-1)/n) * (d/dx)P_n(x) = xP_n(x) - P_n-1(x). This formula is useful for calculating derivatives and holds even at the endpoints of the interval.

Legendre Polynomials Wiki: Legendre Polynomials: Foundations and Frontiers in Applied Mathematics

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Introduction

What are Legendre Polynomials?

Legendre polynomials, named after the French mathematician Adrien-Marie Legendre (1782), constitute a fundamental system of complete and orthogonal polynomials. These mathematical constructs possess a rich array of properties and find extensive utility across various scientific and engineering disciplines. Their definitions can be approached from multiple perspectives, each illuminating distinct characteristics and suggesting connections to diverse mathematical structures and practical applications.

Related Concepts

The study of Legendre polynomials is intertwined with several related mathematical entities. These include:

Associated Legendre Polynomials: A generalization involving two parameters.
Legendre Functions: Solutions to Legendre's differential equation with non-integer parameters.
Legendre Functions of the Second Kind: Another class of non-polynomial solutions to Legendre's differential equation.
Big q-Legendre Polynomials: A q-analog of Legendre polynomials.

Understanding these related concepts provides a broader context for the significance and versatility of Legendre polynomials.

Visualizing the First Few

While direct visualization is not possible here, imagine a graph illustrating the first six Legendre polynomials. These functions exhibit distinct oscillatory behaviors within the interval [-1, 1], with each successive polynomial having one more root than its predecessor. This graphical representation helps in intuitively grasping their orthogonal nature and how they form a basis for functions over this interval.

Definitions

Orthogonal System Construction

One foundational approach defines Legendre polynomials as an orthogonal system. Specifically, a polynomial P_n(x) of degree n is orthogonal with respect to the weight function w(x)=1 over the interval [-1,1]. This means that the integral of the product of two distinct Legendre polynomials over this interval is zero:

\int -1 1 P m (x)P n (x) dx = 0 if n \neq m.

Coupled with the standardization condition P_n(1)=1, each polynomial can be uniquely determined. This systematic construction begins with P₀(x)=1 and P₁(x)=x, with subsequent polynomials derived by ensuring orthogonality to all lower-degree polynomials. This method highlights their fundamental role as one of the three classical orthogonal polynomial systems, alongside Laguerre polynomials (orthogonal over [0,∞) with weight e^-x) and Hermite polynomials (orthogonal over (-∞,∞) with weight e^-x²).

Generating Function Approach

Legendre polynomials can also be defined as the coefficients in a formal power series expansion of a specific generating function:

(1 / \sqrt(1 - 2xt + t²)) = \sum n=0 \infty P n (x)t n .

Here, P_n(x) is the coefficient of tⁿ. This definition is historically significant, as it is how Legendre himself first introduced these polynomials in 1782, directly linking them to the multipole expansion in electrostatics. Differentiating this generating function with respect to t and equating coefficients leads to Bonnet's recursion formula:

(n+1)P n+1 (x) = (2n+1)xP n (x) - nP n-1 (x).

This powerful recurrence relation, combined with P₀(x)=1 and P₁(x)=x, allows for the recursive generation of all higher-order Legendre polynomials without direct series expansion.

Differential Equation Definition

A third crucial definition arises from Legendre's differential equation:

(1-x²)P n''(x) - 2xP n'(x) + n(n+1)P n (x) = 0.

For integer values of n, the solutions to this equation that are regular at x = ±1 are precisely the Legendre polynomials. This perspective is deeply connected to Sturm-Liouville theory, where the differential equation is viewed as an eigenvalue problem. The eigenvalues are of the form n(n+1), and the eigenfunctions are P_n(x), with their orthogonality and completeness naturally following from this theoretical framework.

In physical contexts, Legendre's differential equation frequently appears when solving Laplace's equation (and related partial differential equations) using separation of variables in spherical coordinates. This highlights their intrinsic link to rotational symmetry, as they represent the angular part of the Laplacian operator's eigenfunctions, specifically the subset invariant under rotations about the polar axis.

Rodrigues' Formula and Explicit Forms

Rodrigues' formula offers a remarkably compact and elegant expression for Legendre polynomials:

P n (x) = (1 / (2 n n!)) * (d n /dx n)(x² - 1) n .

This formula is instrumental in deriving many of their properties and explicit representations. For instance, it leads to power series expansions and other forms involving binomial coefficients. The first few Legendre polynomials are:

`n`	`P_n(x)`
0	`1`
1	`x`
2	`(1/2)(3x² - 1)`
3	`(1/2)(5x³ - 3x)`
4	`(1/8)(35x⁴ - 30x² + 3)`
5	`(1/8)(63x⁵ - 70x³ + 15x)`
6	`(1/16)(231x⁶ - 315x⁴ + 105x² - 5)`
7	`(1/16)(429x⁷ - 693x⁵ + 315x³ - 35x)`
8	`(1/128)(6435x⁸ - 12012x⁶ + 6930x⁴ - 1260x² + 35)`
9	`(1/128)(12155x⁹ - 25740x⁷ + 18018x⁵ - 4620x³ + 315x)`
10	`(1/256)(46189x¹⁰ - 109395x⁸ + 90090x⁶ - 30030x⁴ + 3465x² - 63)`

Main Properties

Orthogonality and Normalization

The defining characteristic of Legendre polynomials is their orthogonality over the interval [-1, 1] with respect to a weight function of 1. The standardization P_n(1)=1 further fixes their normalization. Rodrigues' formula allows for the derivation of the normalization integral:

\int -1 1 P n (x)² dx = 2 / (2n+1).

Combining orthogonality and normalization, the relationship can be concisely expressed using the Kronecker delta (δ_mn):

\int -1 1 P m (x)P n (x) dx = (2 / (2n+1)) δ mn .

Completeness

The completeness of Legendre polynomials signifies their ability to represent a broad class of functions. Any piecewise continuous function f(x) with a finite number of discontinuities in [-1, 1] can be approximated by a series of Legendre polynomials:

f n (x) = \sum ℓ=0 n a ℓ P ℓ (x)

This sum converges in the mean to f(x) as n → ∞, provided the coefficients a_ℓ are calculated as:

a ℓ = ((2ℓ+1)/2) \int -1 1 f(x)P ℓ (x) dx.

This property is fundamental to various expansion techniques in mathematical physics and engineering, allowing complex functions to be decomposed into a simpler, orthogonal basis.

Additional Properties

Legendre polynomials exhibit several other useful properties:

Parity: They are either even or odd functions, following the rule: P_n(-x) = (-1)ⁿP_n(x).
Integral Property: For n ≥ 1, the integral over the interval is zero: ∫_-1¹ P_n(x) dx = 0. This implies that the average of any Legendre series over [-1, 1] is simply its leading coefficient, a₀.
Derivative at Endpoint: The derivative at x=1 is given by: P_n'(1) = n(n+1)/2.
Product Expansion: The product of two Legendre polynomials can be expanded as a sum of other Legendre polynomials.
Askey-Gasper Inequality: A significant inequality stating that ∑_j=0ⁿ P_j(x) ≥ 0 for x ≥ -1.
Spherical Harmonics Connection: Legendre polynomials of a scalar product of unit vectors can be expanded using spherical harmonics, revealing their deep connection to angular momentum in quantum mechanics and potential theory.

Recurrence Relations

Beyond Bonnet's formula, other recurrence relations are crucial for manipulating and calculating Legendre polynomials and their derivatives:

(x²-1)/n * (d/dx)P n (x) = xP n (x) - P n-1 (x)

And an alternative expression for derivatives:

(d/dx)P n+1 (x) = (n+1)P n (x) + x(d/dx)P n (x)

For integration, a particularly useful relation is:

(2n+1)P n (x) = (d/dx)(P n+1 (x) - P n-1 (x))

These relations simplify calculations and provide pathways to derive further properties, making them indispensable tools in theoretical and applied mathematics.

Applications

Inverse Distance Potential & Physics

Legendre polynomials were initially conceived by Legendre in 1782 as coefficients in the expansion of the Newtonian potential, 1/|x - x'|. This expression is fundamental to describing gravitational and Coulomb potentials. In spherical coordinates, this expansion takes the form:

(1 / |x - x'|) = (1 / \sqrt(r² + r'² - 2rr'cosγ)) = \sum ℓ=0 \infty (r' ℓ / r ℓ+1) P ℓ (cosγ),

where r and r' are vector magnitudes, and γ is the angle between them. This series converges when r > r'. This application is crucial for integrating potentials over continuous mass or charge distributions. Furthermore, Legendre polynomials naturally arise in the solution of Laplace's equation for static potentials in charge-free regions with axial symmetry, and in solving the Schrödinger equation for central forces in quantum mechanics.

Multipole Expansions

The generating function for Legendre polynomials is directly applicable to multipole expansions, which are used to approximate potentials from charge or mass distributions. For instance, the electric potential Φ(r,θ) due to a point charge on the z-axis at z=a is proportional to 1/R = 1/√(r² + a² - 2arcosθ). If the observation point's radius r is greater than a, the potential can be expanded as:

Φ(r,θ) \propto (1/r) \sum k=0 \infty (a/r) k P k (cosθ).

This expansion forms the basis of the standard multipole expansion. Conversely, if r < a, an "interior multipole expansion" can be derived by exchanging a and r in the formula, demonstrating the versatility of Legendre polynomials in characterizing fields at different distances from a source.

In Trigonometry

Legendre polynomials also find utility in expressing trigonometric functions. For example, Chebyshev polynomials of the first kind, T_n(cosθ) ≡ cos(nθ), can be expanded as linear combinations of Legendre polynomials P_n(cosθ). The first few orders illustrate this relationship:

T 0 (cosθ) = 1 = P 0 (cosθ) T 1 (cosθ) = cosθ = P 1 (cosθ) T 2 (cosθ) = cos(2θ) = (1/3)(4P 2 (cosθ) - P 0 (cosθ)) T 3 (cosθ) = cos(3θ) = (1/5)(8P 3 (cosθ) - 3P 1 (cosθ))

This connection highlights their role in harmonic analysis and signal processing, where such expansions can simplify complex periodic functions. Another interesting trigonometric identity involving Legendre polynomials is:

(sin((n+1)θ) / sinθ) = \sum ℓ=0 n P ℓ (cosθ)P n-ℓ (cosθ).

Recurrent Neural Networks

In the realm of modern computational science, Legendre polynomials have found an unexpected application in recurrent neural networks (RNNs). Specifically, a d-dimensional memory vector m ∈ ℝ^d within an RNN can be optimized such that its neural activities adhere to a linear time-invariant system described by a state-space representation. The past input u(t-θ') over a sliding window of θ units of time can be effectively approximated by a linear combination of the first d shifted Legendre polynomials, weighted by the elements of m(t):

u(t-θ') \approx \sum ℓ=0 d-1 P̃ ℓ (θ'/θ) m ℓ (t), 0 \leq θ' \leq θ.

Networks employing this approach, known as Legendre Memory Units (LMUs), have demonstrated superior performance and computational efficiency compared to traditional architectures like Long Short-Term Memory (LSTM) units, showcasing the enduring relevance of these classical polynomials in cutting-edge fields like deep learning.

Zeros

Location and Properties

A significant property of Legendre polynomials P_n(x) is that all n of their zeros are real, distinct from one another, and confined within the open interval (-1, 1). Furthermore, these zeros exhibit an "interlacing property": if you consider the interval [-1, 1] divided into n+1 subintervals by the zeros of P_n(x), each of these subintervals will contain exactly one zero of P_n+1(x). Due to the parity property (P_n(-x) = (-1)ⁿP_n(x)), if x_k is a zero of P_n(x), then -x_k is also a zero.

Numerical Integration

The precise locations of these zeros are of paramount importance in numerical analysis, particularly in the context of Gaussian quadrature. The specific quadrature method based on the zeros of Legendre polynomials is known as Gauss-Legendre quadrature. This technique provides highly accurate approximations of definite integrals by strategically choosing evaluation points (the zeros) and corresponding weights, making it a cornerstone of computational mathematics.

Distribution and Extrema

The zeros of P_n(cosθ) are distributed nearly uniformly over the range θ ∈ (0,π). More precisely, there is one zero θ in each interval (π(k+1/2)/(n+1/2), π(k+1)/(n+1/2)) for k=0,1,...,n-1. This uniform-like distribution is a consequence of the Dirichlet-Mehler formulas. This property, combined with the fact that P_n(±1) ≠ 0, implies that P_n(x) has exactly n-1 local minima and maxima within the interval (-1, 1). Equivalently, the derivative dP_n(x)/dx has n-1 zeros in (-1, 1).

Variants

Shifted Legendre Polynomials

The shifted Legendre polynomials, denoted as P̃_n(x), are a transformation of the standard Legendre polynomials designed to be orthogonal over the interval [0, 1] instead of [-1, 1]. They are defined by an affine transformation:

P̃ n (x) = P n (2x-1).

This transformation maps [0, 1] bijectively to [-1, 1], ensuring that P̃_n(x) retain the orthogonality property on the new interval:

\int 01 P̃ m (x)P̃ n (x) dx = (1 / (2n+1)) δ mn .

An explicit expression for these polynomials is:

P̃ n (x) = (-1) n \sum k=0 n (n k)(n+k k)(-x) k .

Their Rodrigues' formula analogue is:

P̃ n (x) = (1 / n!) * (d n /dx n)(x² - x) n .

The first few shifted Legendre polynomials are:

`n`	`P̃_n(x)`
0	`1`
1	`2x - 1`
2	`6x² - 6x + 1`
3	`20x³ - 30x² + 12x - 1`
4	`70x⁴ - 140x³ + 90x² - 20x + 1`
5	`252x⁵ - 630x⁴ + 560x³ - 210x² + 30x - 1`

Legendre Rational Functions

Legendre rational functions extend the concept of orthogonality to the interval [0, ∞). These are a sequence of orthogonal functions derived by composing the Cayley transform with Legendre polynomials. A rational Legendre function of degree n is defined as:

R n (x) = (\sqrt2 / (x+1)) * P n ((x-1)/(x+1)).

These functions serve as eigenfunctions for a singular Sturm-Liouville problem, characterized by eigenvalues λ_n = n(n+1). This variant is particularly useful in applications requiring orthogonal bases over semi-infinite domains, such as certain types of spectral methods for solving differential equations.

Related Concepts

Broader Contexts

Legendre polynomials are part of a larger family of orthogonal polynomials and functions that are indispensable in various branches of mathematics and physics. Understanding their connections to these related concepts enriches one's appreciation of their mathematical elegance and practical utility.

Gaussian Quadrature: A numerical integration technique that leverages the zeros of orthogonal polynomials for optimal accuracy.
Gegenbauer Polynomials: A generalization of Legendre polynomials, also known as ultraspherical polynomials.
Turán's Inequalities: A set of inequalities concerning orthogonal polynomials, including Legendre polynomials.
Legendre Wavelet: Wavelets constructed using Legendre polynomials, useful in signal processing and numerical analysis.
Legendre Function: Solutions to Legendre's differential equation for non-integer parameters.
Jacobi Polynomials: Another family of orthogonal polynomials that generalize Legendre and Chebyshev polynomials.
Romanovski Polynomials: A class of orthogonal polynomials related to the generalized hypergeometric function.
Laplace Expansion (Potential): The mathematical framework where Legendre polynomials first emerged, describing potentials in electrostatics and gravitation.

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References

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Legendre Polynomials: Foundations and Frontiers

💡 Dive in with Flashcard Learning! 💡

Introduction 📚

🧐 What are Legendre Polynomials?

🔗 Related Concepts

📊 Visualizing the First Few

Definitions 📝

📐 Orthogonal System Construction

✨ Generating Function Approach

⚙️ Differential Equation Definition

📜 Rodrigues' Formula and Explicit Forms

Main Properties ✨

⊥ Orthogonality and Normalization

♾️ Completeness

➕ Additional Properties

🔄 Recurrence Relations

Applications 💡

⚛️ Inverse Distance Potential & Physics

⚡ Multipole Expansions

〰️ In Trigonometry

🧠 Recurrent Neural Networks

Zeros ✖️

🎯 Location and Properties

🔢 Numerical Integration

📈 Distribution and Extrema

Variants 🔀

↔️ Shifted Legendre Polynomials

➗ Legendre Rational Functions

Related Concepts 🌐

🎓 Broader Contexts

Teacher's Corner 🧑‍🏫

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References

📜 References

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Disclaimer ⚠️

📜 Important Notice

Dive in with Flashcard Learning!

Introduction

What are Legendre Polynomials?

Related Concepts

Visualizing the First Few

Definitions

Orthogonal System Construction

Generating Function Approach

Differential Equation Definition

Rodrigues' Formula and Explicit Forms

Main Properties

Orthogonality and Normalization

Completeness

Additional Properties

Recurrence Relations

Applications

Inverse Distance Potential & Physics

Multipole Expansions

In Trigonometry

Recurrent Neural Networks

Zeros

Location and Properties

Numerical Integration

Distribution and Extrema

Variants

Shifted Legendre Polynomials

Legendre Rational Functions

Related Concepts

Broader Contexts

Teacher's Corner

True or False?

Test Your Knowledge!

Gamer's Corner

Discover other topics to study!

References

Feedback & Support

Disclaimer

Important Notice