Orthogonal Polynomials

Orthogonal polynomials are classes of Polynomials $\{p_n(x)\}$ over a range $[a,b]$ which obey an Orthogonality relation

$$\int_a^b w(x) p_m(x)p_n(x)\,dx=\delta_{mn}c_n,$$ (1)

where $w(x)$ is a Weighting Function and $\delta$ is the Kronecker Delta. If $c_m=1$, then the Polynomials are not only orthogonal, but orthonormal.

Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier Series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important Differential Equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt Orthonormalization. Abramowitz and Stegun (1972, pp. 774-775) give a table of common orthogonal polynomials.

Type	Interval	$w(x)$	$c_n$
Chebyshev Polynomial of the First Kind	$[-1,1]$	$(1-x^2)^{-1/2}$	$\cases{{\textstyle{1\over 2}}\pi & for $n=0$\cr \pi & otherwise\cr}$
Chebyshev Polynomial of the Second Kind	$[-1,1]$	$\sqrt{1-x^2}$	${\textstyle{1\over 2}}\pi$
Hermite Polynomial	$(-\infty,\infty)$	$e^{-x^2}$	$\sqrt{\pi}\,2^n n!$
Jacobi Polynomial	$(-1,1)$	$(1-x)^\alpha(1+x)^\beta$	$h_n$
Laguerre Polynomial	$[0,\infty)$	$e^{-x}$	1
Laguerre Polynomial (Associated)	$[0,\infty)$	$x^ke^{-x}$	${(n+k)!\over n!}$
Legendre Polynomial	$[-1,1]$	1	${2\over 2n+1}$
Ultraspherical Polynomial	$[-1,1]$	$(1-x^2)^{\alpha-1/2}$	$\cases{ {2^{1-2\alpha}\pi\Gamma(n+2\alpha)\over n!(n+\alpha)[\Gamma(\alpha)]^2} & for $\alpha\not=0$\cr {2\pi\over n^2} & for $\alpha=0$.\cr}$

In the above table, the normalization constant is the value of

$$c_n\equiv \int w(x)[p_n(x)]^2\,dx$$ (2)

and

$$h_n\equiv {2^{\alpha+\beta+1}\over 2n+\alpha+\beta+1} {\Gamma(n+\alpha+1)\Gamma(n+\beta+1)\over n!\Gamma(n+\alpha+\beta+1)},$$ (3)

where $\Gamma(z)$ is a Gamma Function.

The Roots of orthogonal polynomials possess many rather surprising and useful properties. For instance, let $x_1<x_2<\ldots<x_n$ be the Roots of the $p_n(x)$ with $x_0=a$ and $x_{n+1}=b$. Then each interval $[x_\nu, x_{\nu+1}]$ for $\nu=0$, 1, ..., $n$ contains exactly one Root of $p_{n+1}(x)$. Between two Roots of $p_n(x)$ there is at least one Root of $p_m(x)$ for $m>n$.

Let $c$ be an arbitrary Real constant, then the Polynomial

$$p_{n+1}(x)-cp_n(x)$$ (4)

has $n+1$ distinct Real Roots. If $c>0$ ($c<0$), these Roots lie in the interior of $[a,b]$, with the exception of the greatest (least) Root which lies in $[a,b]$ only for

$$c\leq {p_{n+1}(b)\over p_n(b)}\qquad \left({c\geq{p_{n+1}(a)\over p_n(a)}}\right).$$ (5)

The following decomposition into partial fractions holds

$${p_n(x)\over p_{n+1}(x)} =\sum_{\nu=0}^n {l_\nu\over x-\xi},$$ (6)

where $\{\xi_\nu\}$ are the Roots of $p_{n+1}(x)$ and

$$l_\nu = {p_n(\xi_\nu)\over p_{n+1}'(\xi_\nu)}$$

$$= {p_{n+1}'(\xi_\nu)p_n(\xi_\nu)-p_n'(\xi_\nu)'p_{n+1}(\xi_\nu)\over[p_{n+1}'(\xi_\nu)]^2}>0.$$ (7)

Another interesting property is obtained by letting $\{p_n(x)\}$ be the orthonormal set of Polynomials associated with the distribution $d\alpha(x)$ on $[a,b]$. Then the Convergents $R_n/S_n$ of the Continued Fraction

$${1\over A_1x+B_1}-{C_2\over A_2x+B_2}-{C_3\over A_3x+B_3}-\ldots-{C_n\over A_nx+B_n}+\ldots$$ (8)

are given by

$$R_n = R_n(x)$$

$$= {c_0}^{-3/2}\sqrt{c_0c_2-{c_1}^2}\int_a^b {p_n(x)-p_n(t)\over x-t} \,d\alpha(t)$$ (9)

$$S_n = S_n(x)=\sqrt{c_0}\,p_n(x),$$ (10)

where $n=0$, 1, ...and

$$c_n=\int_a^b x^n\,d\alpha(x).$$ (11)

Furthermore, the Roots of the orthogonal polynomials $p_n(x)$ associated with the distribution $d\alpha(x)$ on the interval $[a,b]$ are Real and distinct and are located in the interior of the interval $[a,b]$.

See also Chebyshev Polynomial of the First Kind, Chebyshev Polynomial of the Second Kind, Gram-Schmidt Orthonormalization, Hermite Polynomial, Jacobi Polynomial, Krawtchouk Polynomial, Laguerre Polynomial, Legendre Polynomial, Orthogonal Functions, Spherical Harmonic, Ultraspherical Polynomial, Zernike Polynomial

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. ``Orthogonal Polynomials.'' Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 520-521, 1985.

Iyanaga, S. and Kawada, Y. (Eds.). ``Systems of Orthogonal Functions.'' Appendix A, Table 20 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1477, 1980.

Nikiforov, A. F.; Uvarov, V. B.; and Suslov, S. S. Classical Orthogonal Polynomials of a Discrete Variable. New York: Springer-Verlag, 1992.

Sansone, G. Orthogonal Functions. New York: Dover, 1991.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 44-47 and 54-55, 1975.