Problem 3.9.1. Using Rodrigues’ formula, compute \(P_0,\dots ,P_4\) and verify that \(P_n(1)=1\) and \(P_n(-1)=(-1)^{n}\) in each case. Where to start: differentiate \((x^{2}-1)^{n}\) the required number of times; for the endpoint values, note that \((x^{2}-1)^{n}=(x-1)^{n}(x+1)^{n}\) and only one term survives at \(x=1\).

Problem 3.9.2. Prove the Legendre recurrence \[(n+1)P_{n+1}(x) = (2n+1)x\,P_n(x) - n\,P_{n-1}(x)\] from the generating function \(\left (1-2xt+t^{2}\right )^{-1/2} = \sum P_n(x)t^{n}\). Where to start: differentiate the generating function with respect to \(t\), multiply through by \(1-2xt+t^{2}\), and compare coefficients of \(t^{n}\).

Problem 3.9.3. Show that \[\int ^{1}_{-1}P_m(x)P_n(x)\,dx = \frac {2}{2n+1}\,\delta _{mn}.\] Obtain the orthogonality from Theorem 4.3.2 rather than by direct integration, and the normalisation by any method you prefer.

Problem 3.9.4. Expand \(f(x)=|x|\) on \([-1,1]\) in Legendre polynomials as far as the term in \(P_4\), and explain why every odd coefficient vanishes. Where to start: \(|x|\) is even and \(P_n\) has parity \((-1)^{n}\), so the integrand is odd whenever \(n\) is odd.

Problem 3.9.5. State Hermite’s differential equation in the convention used in these notes, and verify that \(H_n\) as defined by the Rodrigues formula of Section 3.6 satisfies it.

Problem 3.9.6. These notes use the probabilists’ Hermite polynomials, \[H_n(x) = (-1)^{n}e^{x^{2}/2}\frac {d^{n}}{dx^{n}}e^{-x^{2}/2},\] with generating function \(\exp \left (xt-t^{2}/2\right )\) and weight \(e^{-x^{2}/2}\). Much of the literature uses the physicists’ polynomials \(\widetilde {H}_n\), defined with \(e^{-x^{2}}\) throughout, with generating function \(\exp \left (2xt-t^{2}\right )\) and weight \(e^{-x^{2}}\).

(a)

Show that \(\widetilde {H}_n(x) = 2^{n/2}H_n\!\left (x\sqrt 2\right )\).

(b)

Deduce that \(\displaystyle \int ^{\infty }_{-\infty }e^{-x^{2}}\widetilde {H}_m(x)\widetilde {H}_n(x)\,dx = 2^{n}n!\sqrt {\pi }\,\delta _{mn}\), and find the corresponding norm for \(H_n\) under weight \(e^{-x^{2}/2}\).

(c)

State the eigenvalue of each in its Sturm–Liouville form.

Where to start: substitute \(x\mapsto x\sqrt 2\) in one generating function and compare with the other.

Problem 3.9.7. Prove that the Laguerre polynomials satisfy \[\int ^{\infty }_{0}x^{\alpha }e^{-x}L^{\alpha }_m(x)L^{\alpha }_n(x)\,dx = \frac {\Gamma (n+\alpha +1)}{n!}\,\delta _{mn}.\] Where to start: orthogonality follows from Theorem 4.3.2 with \(w(x)=x^{\alpha }e^{-x}\); for the norm, use the Rodrigues formula and integrate by parts \(n\) times.

Problem 3.9.8. Show that the Chebyshev polynomials satisfy \(T_n(\cos \theta )=\cos n\theta \), and use this to prove the recurrence \(T_{n+1}(x) = 2x\,T_n(x)-T_{n-1}(x)\) and the equal-ripple property visible in Figure 3.2. Where to start: the recurrence is the identity \(\cos (n+1)\theta + \cos (n-1)\theta = 2\cos \theta \cos n\theta \).

Problem 3.9.9. Show that \(T_n\) has \(n\) distinct zeros in \((-1,1)\), at \(x_k = \cos \left [\left (2k-1\right )\pi /(2n)\right ]\) for \(k=1,\dots ,n\). These are the Chebyshev nodes; explain briefly why interpolating at them is preferable to using equally spaced points.

Problem 3.9.10. Verify that the Jacobi polynomials reduce to the Legendre polynomials when \(\alpha =\beta =0\), and to the Chebyshev polynomials of the first and second kinds when \(\alpha =\beta =-\tfrac 12\) and \(\alpha =\beta =\tfrac 12\) respectively, up to normalisation. Where to start: compare the weight functions in Table 4.1, then check the eigenvalues agree.