Question

The origin is an ordinary point of the Chebyshev equation, $$ \left(1-z^{2}\right) y^{\prime \prime}-z y^{\prime}+m^{2} y=0 $$ which therefore has series solutions of the form \(z^{\sigma} \sum_{0}^{\infty} a_{n} z^{n}\) for \(\sigma=0\) and \(\sigma=1\). (a) Find the recurrence relationships for the \(a_{n}\) in the two cases and show that there exist polynomial solutions \(T_{m}(z)\) : (i) for \(\sigma=0\), when \(m\) is an even integer, the polynomial having \(\frac{1}{2}(m+2)\) terms; (ii) for \(\sigma=1\), when \(m\) is an odd integer, the polynomial having \(\frac{1}{2}(m+1)\) terms. (b) \(T_{m}(z)\) is normalised so as to have \(T_{m}(1)=1 .\) Find explicit forms for \(T_{m}(z)\) for \(m=0,1,2,3\). (c) Show that the corresponding non-terminating series solutions \(S_{m}(z)\) have as their first few terms $$ \begin{aligned} &S_{0}(z)=a_{0}\left(z+\frac{1}{3 !} z^{3}+\frac{9}{5 !} z^{5}+\cdots\right) \\\ &S_{1}(z)=a_{0}\left(1-\frac{1}{2 !} z^{2}-\frac{3}{4 !} z^{4}-\cdots\right) \\\ &S_{2}(z)=a_{0}\left(z-\frac{3}{3 !} z^{3}-\frac{15}{5 !} z^{5}-\cdots\right) \\\ &S_{3}(z)=a_{0}\left(1-\frac{9}{2 !} z^{2}+\frac{45}{4 !} z^{4}+\cdots\right) \end{aligned} $$

Short answer

For \( \theta = 0 \) and even \( m \), the recurrence relation is \( a_{n+2} = \frac{n^{2} - m^{2}}{(n+1)(n+2)} a_{n} \). For odd \( m \) and \( \theta = 1 \), the recurrence relation is \( a_{n+2} = \frac{(n+1)^{2} - m^{2}}{(n+2)(n+3)} a_{n} \).

Step-by-step solution

Form of the Chebyshev Differential Equation

Consider the Chebyshev differential equation given by: \[ (1 - z^{2}) y'' - z y' + m^{2} y = 0 \]The goal is to find the series solutions of the form \( y = z^{\theta} \sum_{0}^{\text{infty}} a_{n} \ z^{n} \) for \( \theta = 0 \) and \( \theta = 1 \).

Series Solution for \( \theta = 0 \)

Assume \( \theta = 0 \), so the series solution is:\[ y = \sum_{0}^{\infty} a_{n} z^{n} \]Compute the derivatives: \[ y' = \sum_{1}^{\infty} n a_{n} z^{n-1} \]\[ y'' = \sum_{2}^{\infty} n(n-1) a_{n} z^{n-2} \]Substitute into the differential equation.

Substitute and Collect Terms for \( \theta = 0 \)

Plugging the series expressions into the differential equation:\[ (1 - z^{2}) \sum_{2}^{\infty} n(n-1) a_{n} z^{n-2} - z \sum_{1}^{\infty} n a_{n} z^{n-1} + m^{2} \sum_{0}^{\infty} a_{n} z^{n} = 0 \]Combine like terms to form a single series.

Recurrence Relation for \( \theta = 0 \)

Set the coefficient of each power of \( z \) to zero to obtain the recurrence relation:\[ a_{n+2} = \frac{n^{2} - m^{2}}{(n+1)(n+2)} a_{n} \]

Polynomial Solutions T_m(z) for Even m (\( \theta = 0 \))

For even \( m \), the series terminates, resulting in a polynomial. For example, if \( m = 2 \), the recurrence yields only terms up to \( z^{2} \). Thus, \( T_{m}(z) \) has \( \frac{m+2}{2} \) terms.

Series Solution for \( \theta = 1 \)

Assume \( \theta = 1 \), so the series solution is:\[ y = z \sum_{0}^{\infty} a_{n} z^{n} \]Compute the derivatives similarly and substitute into the differential equation.

Substitute and Collect Terms for \( \theta = 1 \)

Plugging the series expressions into the differential equation:\[ (1 - z^{2}) \sum_{0}^{\infty} (n+1)(n+2) a_{n+1} z^{n} - z \sum_{0}^{\infty} (n+1) a_{n} z^{n} + m^{2} \sum_{0}^{\infty} a_{n} z^{n+1} = 0 \]Combine terms to form a single series in \( z^{n} \).

Recurrence Relation for \( \theta = 1 \)

Set coefficients to zero to obtain the recurrence relation:\[ a_{n+2} = \frac{(n+1)^{2} - m^{2}}{(n+2)(n+3)} a_{n} \]

Polynomial Solutions T_m(z) for Odd m (\( \theta = 1 \))

For odd \( m \), the series terminates, yielding a polynomial. For example, if \( m = 3 \), the recurrence yields only terms up to \( z^{3} \). Thus, \( T_{m}(z) \) has \( \frac{m+1}{2} \) terms.

Normalized Polynomials T_m(1)

Find explicit forms for \( T_{m}(z) \) for \( m = 0, 1, 2, 3 \). Normalize such that \( T_{m}(1) = 1 \).

First Few Terms of Non-Terminating Series Solutions

Use the recurrence relations to write the first few terms of the non-terminating series solutions for \( S_{0}(z) \), \( S_{1}(z) \), \( S_{2}(z) \) and \( S_{3}(z) \) given.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving functions and their derivatives. They describe various phenomena like motion, growth, decay, etc. In the Chebyshev Differential Equation \((1 - z^{2}) y'' - z y' + m^{2} y = 0\), the function we seek is y, and it depends on the variable z. The goal is to find solutions y that satisfy this equation. Understanding how functions change, through their derivatives, allows us to model real-world systems mathematically.

Series Solutions

A common method to solve ODEs is by finding series solutions. This involves expressing the solution as an infinite sum of terms. For example, for \( \theta = 0\), we assume the series solution \[ y = \sum_{0}^{\text{∞}} a_{n} z^{n} \]. We then compute the derivatives and substitute them into the original differential equation. This helps us find a pattern in the coefficients \(a_{n}\), known as a recurrence relation.
By solving these relations, we can express the function y as a power series. This is especially useful when dealing with complex differential equations that are difficult to solve directly.

Recurrence Relations

Recurrence relations are equations that define sequences of numbers using preceding terms. They play a crucial role in calculating coefficients in series solutions. For instance, substituting the series into the Chebyshev Differential Equation and equating the coefficients of like powers of z helps derive recurrence relations for both \( \theta = 0\) and \( \theta = 1\).

• When \( \theta = 0\), the relation is \[ a_{n+2} = \frac{n^{2} - m^{2}}{(n+1)(n+2)} a_{n} \].
• When \( \theta = 1\), it is \[ a_{n+2} = \frac{(n+1)^{2} - m^{2}}{(n+2)(n+3)} a_{n} \].

Solving these allows us to identify patterns in the sequence and thus find the coefficients of each term in the power series.

Chebyshev Polynomials

Chebyshev Polynomials are a set of orthogonal polynomials arising in the solutions to the Chebyshev Differential Equation. They are denoted by \( T_{m}(z)\). For even m, the polynomial has \[ \frac{m+2}{2} \] terms, and for odd m, it has \[ \frac{m+1}{2} \] terms. These polynomials are normalized such that \( T_{m}(1) = 1\). Examples include:

• \( T_{0}(z) \) is simply 1
• \( T_{1}(z) = z \)
• \( T_{2}(z) = 2z^{2} - 1 \)
• \( T_{3}(z) = 4z^{3} - 3z \)

Chebyshev Polynomials are fundamental in numerical analysis and approximation theory, helping minimize the error in polynomial approximations.