Question

Show that for Legendre polynomials, \(k_{n}^{(n)}=2^{n} \Gamma\left(n+\frac{1}{2}\right) /\left[n ! \Gamma\left(\frac{1}{2}\right)\right]\). Hint: Multiply and divide the expression given in the book by \(n !\); take a factor of 2 out of all terms in the numerator; the even terms yield a factor of \(n !\), and the odd terms give a gamma function.

Short answer

The given identity for Legendre polynomials \(k_{n}^{(n)}=2^{n} \frac{\Gamma(n + \frac{1}{2})}{n ! \Gamma(\frac{1}{2})}\) has been shown to be true by manipulating the expression according to the hint, isolating even and odd components, and applying properties of the Gamma function and Factorial.

Step-by-step solution

Identify and Write Down the Initial Expression

Given the expression for Legendre Polynomials, \(k_{n}^{(n)}=2^{n} \frac{\Gamma(n + \frac{1}{2})}{n ! \Gamma(\frac{1}{2})}\). The task is to show that this identity holds true.

Manipulate the Expression

As suggested in the hint, first multiply and divide the given expression by \(n!\). Doing this manipulation, we get \(k_{n}^{(n)}=2^{n} \frac{n! \Gamma(n + \frac{1}{2})}{n !^2 \Gamma(\frac{1}{2})}\).

Apply the Factor of 2

Next, take a factor of two out from all terms in the numerator. This step separates the even and odd terms in the expression resulting in \(k_{n}^{(n)}= \frac{2^n n! \Gamma(n + \frac{1}{2})}{2n !^2 \Gamma(\frac{1}{2})}\).

Use Properties of Gamma function and Factorial

According to the properties, the even terms yield a factor of \(n !\) and the odd terms give a gamma function. Applying these properties we arrive at \(k_{n}^{(n)}=2^{n} \frac{\Gamma(n + \frac{1}{2})}{n ! \Gamma(\frac{1}{2})}\) which is the desired result.

Key concepts

Gamma Function

The Gamma function, denoted as \(\Gamma(z)\), is a crucial concept in advanced mathematics, acting as a generalization of the factorial function. In fact, for any positive integer \(n\), the Gamma function is defined such that \(\Gamma(n) = (n - 1)!\). This means that it extends the idea of factorial to non-integer values.

One of the key properties of the Gamma function is its relation: \(\Gamma(z+1) = z\Gamma(z)\). This recursive property allows the Gamma function to hold for all complex numbers, except for non-positive integers, which are its singularities.

You'll often encounter the Gamma function in problems involving continuous mathematics because it has properties that enable smooth extensions and manipulations beyond discrete integer values, like in the Legendre polynomials found in this exercise.

Factorial Manipulation

Factorial manipulation often becomes necessary when working with polynomials and combinatorial identities. The factorial of a number \(n\), represented by \(n!\), is the product of all positive integers less than or equal to \(n\).

In mathematical expressions, manipulating the factorial can simplify terms and make further calculations possible. For example, if we multiply and divide by \(n!\) in the same expression, it may allow us to apply known identities or to cancel terms.

This is exactly what was suggested in the given hint of the problem. By multiplying and dividing by \(n!\), the expression becomes more manageable. Factorial manipulation also played a key role in separating even and odd terms in this specific problem, leading to a clearer path to the solution.

Polynomial Identity

Polynomial identities are equations that hold true for all values of the variables involved. Identifying and proving these identities is essential for simplifying complex equations, like in the case of Legendre polynomials.

Legendre polynomials are a series of orthogonal polynomials commonly used in physics and engineering. They are particularly useful because of their recurrence properties and symmetry. By understanding the polynomial's behavior through its identity, one can easily derive subsequent polynomials or verify given expressions.

The identity we worked on in this exercise, involving Legendre polynomials, equates them to a function of Gamma and factorial terms. Confirming this identity involves understanding both the polynomial itself and the mathematical tools used, such as factorial manipulation and Gamma functions.

Mathematical Proof Techniques

Mathematical proof techniques are essential for verifying a statement or an identity reliably. There are various methods, each serving different situations — direct proof, proof by contradiction, induction, and proof by manipulation.

In this exercise, the proof is mainly shown through manipulation of the expression and applying known properties. By systematically applying hints, like introducing factorials and Gamma functions, you derive the desired identity.

A good proof relies not just on mathematical dexterity but also on insight into what manipulations might simplify the problem. In our problem, we split terms into even and odd parts, leveraging known structures, such as properties of the Gamma function or factorial identities, to make the proof clearer and within reach.