Abstract
Classical orthogonal polynomials are an important class of special functions. They are intimately related to many of the problems we have discussed in previous chapters. In particular, in many cases they represent eigenfunctions of differential and integral operators.
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Notes
- 1.
Sometimes it is convenient to refer to it as a standard form.
- 2.
Special care is to be taken with the upper and lower bounds of the sums.
- 3.
A finite real \(q_0\) would spawn more general polynomials, called the associated Laguerre polynomials \(L_n^{\alpha }(x)\) with \(\alpha =q_0\), see Problem 4.4. The respective weight function \(w(x)=x^{\alpha }e^{-\frac{Q}{p_1}\;x}\) immediately reduces to (4.15) at \(q_0=0\), so that \(L_n^0(x)=L_n(x)\). These polynomials satisfy the following differential equation:
$$\begin{aligned} xy'' +(\alpha +1-x) y' + ny=0. \end{aligned}$$(4.14) - 4.
As before we want \(\int _a^b dx \, y_n^2(x) w(x) = 1\). Alternatively we set the coefficient in front of highest exponent term \(x^n\) equal to \(1\).
- 5.
One distinguishes the Chebyshev polynomials of the second kind denoted by \(U_n(x)\). They satisfy the equation
$$\begin{aligned} (1-x^2) y'' - 3 x y' + n^2 y = 0, \end{aligned}$$obey the weight function \(w(x) = \sqrt{1-x^2}\) and, consequently, reduce to the Jacobi polynomials \(P_n^{(1/2,\;1/2)}(x)\).
- 6.
The situation with \(r=0\) is simple—this term does not contribute as its integral corresponds to the scalar product of \(y_m\) and \(y_0\), which is equal \(\delta _{m0}N_m\) by definition.
- 7.
Just set the coefficient in front of \(x\) to unity, \(L_1(x) = 1 - x \rightarrow x-1\).
- 8.
A familiar use of this generating function for the Legendre polynomials is to compute the multipole expansion useful in classical field theory or electrostatics. An expansion of the reciprocal distance between two charges situated at the points \(\mathbf{r}_1\) and \(\mathbf{r}_2\) is, see e. g. [27],
$$\begin{aligned} \frac{1}{|\mathbf{r}_1 - \mathbf{r}_2|} = \frac{1}{\sqrt{r_1^2 - 2 r_1 r_2 \cos {\theta } + r_2^2}} = \frac{1}{r_1} \sum _{n=0}^{\infty } P_n(\cos {\theta }) \Bigl (\frac{r_2}{r_1}\Bigr )^n,\;\mathrm{for}\;r_1 > r_2. \end{aligned}$$ - 9.
The same result readily follows from the Rodrigues’ formula (4.25). Indeed, in the limit \(x \rightarrow 1\) we have
$$\begin{aligned} \frac{d^n}{dx^n} (x^2-1)^n = \frac{d^n}{dx^n}\Bigl ((x-1)^n (x+1)^n\Bigr ) \;\rightarrow \;\frac{d^n}{dx^n}(x+1)^n\Bigr |_{x \rightarrow 1} = n!(1+1)^n. \end{aligned}$$ - 10.
Evidently, the same result holds for the canonical form (4.28).
- 11.
The choice of a branch is not important. The right hand-side of the formula (4.47) contains no odd degrees \(\sqrt{x^2-1}\).
- 12.
When \(n\) is non-integer, a solution of the Legendre equation may still be given by the Schl\(\ddot{\mathrm{a}}\)fli integral. The latter defines then a regular function of \(z\) (\({\text {Re}}{z} > - 1\)) in the complex plane with the branch cut from \(-1\) to \(-\infty \). This function is usually called the Legendre function of the first kind and denoted by \(P_n(z)\). Indeed, the function (4.48) has now three branch
points at \(t=z\) and \(t = \pm 1\). Taking the branch cut from \(-1\) to \(-\infty \) and assuming that it does not contain the point \(z\), the suitable contour \(C\) may begin on the real axis at any point \( t> 1\) and must enclose the points \(t=z\) and \(t = 1\). Then the term \((t-z)^{-n-2}\) acquires the factor \(e^{2\pi i(-n-2)}\), the term \((t^2 - 1)^{n+1}\) acquires the factor \(e^{2\pi i(n+1)}\), so that the function (4.48) returns to the original value: \((t^2 - 1)^{n+1}(t-z)^{-n-2} \rightarrow e^{-2\pi i}(t^2 - 1)^{n+1}(t-z)^{-n-2} \equiv (t^2 - 1)^{n+1}(t-z)^{-n-2}\). For further details see [3].
- 13.
For the considered here nonclassical form of the Hermite polynomials the differential Eq. (4.44) reads
$$\begin{aligned} \frac{d H_n(x)}{dx}=-2nH_{n-1}(x). \end{aligned}$$ - 14.
We assume the interval \([a,b]=[-1,1]\).
- 15.
For an arbitrary \(n=\nu \) (and the complex variable \(z\) as well) this equality serves as a definition of the Legendre function\(P_{\nu }(z)\) which we have already encountered on page 211. It is absolutely convergent inside a circle of radius 2 with a center at the point \(z=1\). Here we can draw an analogy to the theory of Gamma function, which is an extension of the factorial function to real and complex numbers.
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Gogolin, A.O., Tsitsishvili, E.G., Komnik, A. (2014). Orthogonal Polynomials. In: Tsitsishvili, E., Komnik, A. (eds) Lectures on Complex Integration. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00212-5_4
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DOI: https://doi.org/10.1007/978-3-319-00212-5_4
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