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A Crash Introduction to Orthogonal Polynomials

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A Crash Introduction to Orthogonal Polynomials A crash introduction to orthogonal polynomials Pavel Sˇˇtov´ıˇcek Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in Prague, Czech Republic Introduction The roots of the theory of orthogonal polynomials go back as far as to the end of the 18th century. The field of orthogonal polynomials was developed to considerable depths in the late 19th century from a study of continued fractions by P. L. Chebyshev, T. J. Stieltjes and others. Some of the mathematicians who have worked on orthogonal polynomials include Hans Ludwig Hamburger, Rolf Herman Nevanlinna, G´abor Szeg}o,Naum Akhiezer, Arthur Erd´elyi,Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al- Salam, and Richard Askey. The theory of orthogonal polynomials is connected with many other branches of mathematics. Selecting a few examples one can mention continued fractions, operator theory (Jacobi operators), moment problems, approximation theory and quadrature, stochastic processes (birth and death processes) and special functions. Some biographical data as well as various portraits of mathematicians are taken from Wikipedia, the free encyclopedia, starting from the web page • http://en.wikipedia.org/wiki/Orthogonal polynomials Most of the theoretical material has been adopted from the fundamental mono- graphs due to Akhiezer and Chihara (detailed references are given below in the text). Classical orthogonal polynomials A scheme of classical orthogonal polynomials • the Hermite polynomials • the Laguerre polynomials, the generalized (associated) Laguerre polynomials • the Jacobi polynomials, their special cases: { the Gegenbauer polynomials, particularly: ∗ the Chebyshev polynomials ∗ the Legendre polynomials 1 Some common features ~ In each case, the respective sequence of orthogonal polynomials, fPn(x); n ≥ 0g, represents an orthogonal basis in a Hilbert space of the type H = L2(I;%(x)dx) where I ⊂ R is an open interval, %(x) > 0 is a continuous function on I. ~ Any sequence of classical orthogonal polynomials fPn(x)g, after having been nor- malized to a sequence of monic polynomials fPn(x)g, obeys a recurrence relation of the type Pn+1(x) = (x − cn)Pn(x) − dnPn−1(x); n ≥ 0; with P0(x) = 1 and where we conventionally put P−1(x) = 0. Moreover, the coef- ficients cn, n ≥ 0, are all real and the coefficients dn, n ≥ 1, are all positive (d0 is arbitrary). The zeros of Pn(x) are real and simple and belong all to I, the zeros of Pn(x) and Pn+1(x) interlace, the union of the zeros of Pn(x) for all n ≥ 0 is a dense subset in I. 8 > R for the Hermite polynomials <> I = (0; +1) for the generalized Laguerre polynomials > :> (−1; 1) for the Jacobi (and Gegenbauer, Chebyshev, Legendre) polynomials Hermite polynomials Charles Hermite: December 24, 1822 { January 14, 1901 References • C. Hermite: Sur un nouveau d´eveloppement en s´eriede fonctions, Comptes Rendus des S´eancesde l'Acad´emiedes Sciences. Paris 58 (1864) 93-100 2 • P.L. Chebyshev: Sur le d´eveloppement des fonctions `aune seule variable, Bulletin physico-math´ematiquede l'Acad´emieImp´erialedes sciences de St.-P´etersbourgh I (1859) 193-200 • P. Laplace: M´emoire sur les int´egrales d´efinieset leur application aux probabilit´es, M´emoiresde la Classe des sciences, math´ematiques et physiques de l'Institut de France 58 (1810) 279-347 Definition (n = 0; 1; 2;:::) bn=2c X (−1)k H (x) = n! (2x)n−2k n k!(n − 2k)! k=0 The Rodrigues formula n n 2 d 2 d H (x) = (−1)nex e−x = 2x − · 1 n dxn dx Orthogonality 1 −x2 p n Hm(x)Hn(x) e dx = π 2 n! δm;n ˆ−∞ 2 The Hermite polynomials form an orthogonal basis of H = L2(R; e−x dx). Recurrence relation Hn+1(x) = 2xHn(x) − 2nHn−1(x); n ≥ 0; H0(x) = 1 and, by convention, H−1(x) = 0. Differential equation The Hermite polynomial Hn(x) is a solution of Hermite's differential equation y00 − 2xy0 + 2ny = 0: 3 Laguerre polynomials and generalized (associated) Laguerre polynomials Edmond Laguerre: April 9, 1834 { August 14, 1886 References 1 e−x • E. Laguerre: Sur l'int´egrale x x dx, Bulletin de la Soci´et´eMath´ematiquede France 7 (1879) 72-81 ´ • N. Y. Sonine: Recherches sur les fonctions cylindriques et le d´eveloppement des fonctions continues en s´eries, Math. Ann. 16 (1880) 1-80 Definition (n = 0; 1; 2;:::) (0) Ln(x) ≡ Ln (x), n n X n (−1)k X n + αxk L (x) = xk;L(α)(x) = (−1)k n k k! n n − k k! k=0 k=0 The Rodrigues formula ex dn 1 d n L (x)= e−xxn = − 1 xn n n! dxn n! dx x−αex dn x−α d n L(α)(x) = e−xxn+α = − 1 xn+α n n! dxn n! dx Orthogonality (α > −1) 1 −x Lm(x)Ln(x) e dx = δm;n ˆ0 1 (α) (α) α −x Γ(n + α + 1) Lm (x)Ln (x) x e dx = δm;n ˆ0 n! 4 The Laguerre polynomials form an orthonormal basis of H = L2((0; 1); e−xdx), the generalized Laguerre polynomials form an orthogonal basis of H = L2((0; 1); xαe−xdx). Recurrence relation (n + 1)Ln+1(x) = (2n + 1 − x)Ln(x) − nLn−1(x); n ≥ 0; more generally, (α) (α) (α) (n + 1)Ln+1(x) = (2n + α + 1 − x) Ln−1(x) − (n + α) Ln−1(x); n ≥ 0; (α) (α) L0 (x) = 1 and, by convention, L−1(x) = 0. Differential equation Ln(x) is a solution of Laguerre's equation x y00 + (1 − x) y0 + n y = 0; (α) more generally, Ln (x) is a solution of the second order differential equation x y00 + (α + 1 − x) y0 + n y = 0: Jacobi (hypergeometric) polynomials Carl Gustav Jacob Jacobi: December 10, 1804 { 18 February 18, 1851 5 References • C.G.J. Jacobi: Untersuchungen ¨uber die Differentialgleichung der hypergeometrischen Reihe, J. Reine Angew. Math. 56 (1859) 149-165 Definition (n = 0; 1; 2;:::) n m Γ(α + n + 1) X n Γ(α + β + n + m + 1) z − 1 P (α,β)(z) = n n! Γ(α + β + n + 1) m Γ(α + m + 1) 2 m=0 The Rodrigues formula (−1)n dn P (α,β)(z) = (1 − z)−α(1 + z)−β (1 − z)α(1 + z)β(1 − z2)n n 2nn! dzn Orthogonality (α; β > −1) 1 α+β+1 (α,β) (α,β) α β 2 Γ(n + α + 1) Γ(n + β + 1) Pm (x)Pn (x) (1 − x) (1 + x) dx = δm;n ˆ−1 (2n + α + β + 1) Γ(n + α + β + 1) n! The Jacobi polynomials form an orthogonal basis of H = L2((−1; 1); (1 − x)α(1 + x)βdx). Recurrence relation (α,β) 2(n + 1)(n + α + β + 1)(2n + α + β) Pn+1 (z) 2 2 (α,β) = (2n + α + β + 1) (2n + α + β + 2)(2n + α + β) z + α − β Pn (z) (α,β) −2(n + α)(n + β)(2n + α + β + 2) Pn−1 (z) ; n ≥ 0; (α,β) (α,β) P0 (z) = 1 and, by convention, P−1 (z) = 0. Differential equation (α,β) The Jacobi polynomial Pn is a solution of the second order differential equation (1 − x2)y00 + (β − α − (α + β + 2) x)y0 + n(n + α + β + 1)y = 0: 6 Gegenbauer (ultraspherical) polynomials Leopold Bernhard Gegenbauer: February 2, 1849 { June 3, 1903 References • L. Gegenbauer: Uber¨ einige bestimmte Integrale, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematische-Naturwissenschaftliche Classe. Wien 70 (1875) 433-443 • L. Gegenbauer: Uber¨ einige bestimmte Integrale, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematische-Naturwissenschaftliche Classe. Wien 72 (1876) 343-354 ¨ ν • L. Gegenbauer: Uber die Functionen Cn (x), Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften. Mathematische-Naturwissenschaftliche Classe. Wien 75 (1877) 891-905 Definition (n = 0; 1; 2;:::) bn=2c X Γ(n − k + α) C(α)(z) = (−1)k (2z)n−2k n Γ(α)k!(n − 2k)! k=0 The Gegenbauer polynomials are a particular case of the Jacobi polynomials Γ(α + 1=2)Γ(2α + n) C(α)(z) = P (α−1=2,α−1=2)(z) n Γ(2α)Γ(n + α + 1=2) n The Rodrigues formula (−2)n Γ(n + α)Γ(n + 2α) dn C(α)(z) = (1 − x2)−α+1=2 (1 − x2)n+α−1=2 n n! Γ(α)Γ(2n + 2α) dxn 7 Orthogonality (α; β > −1) 1 1−2α (α) (α) 2 α−1=2 π2 Γ(n + 2α) Cm (x)Cn (x) (1 − x ) dx = 2 δm;n ˆ−1 (n + α) n! Γ(α) The Gegenbauer polynomials form an orthogonal basis of H = L2((−1; 1); (1 − x2)α−1=2dx). Recurrence relation (α) (α) (α) (n + 1)Cn+1(x) = 2x(n + α)Cn (x) − (n + 2α − 1)Cn−1(x); n ≥ 0; (α) (α) C0 (x) = 1 and, by convention, C−1 (x) = 0. Differential equation Gegenbauer polynomials are solutions of the Gegenbauer differential equation (1 − x2)y00 − (2α + 1)xy0 + n(n + 2α)y = 0: Chebyshev polynomials of the first and second kind Alternative transliterations: Tchebycheff, Tchebyshev, Tschebyschow Pafnuty Lvovich Chebyshev: May 16, 1821 { December 8, 1894 References • P. L. Chebyshev: Th´eoriedes m´ecanismes connus sous le nom de parall´elogrammes, M´emoiresdes Savants ´etrangers pr´esent´es `al'Acad´emiede Saint-P´etersbourg 7 (1854) 539{586 8 Definition (n = 0; 1; 2;:::) T0(x) = 1, U0(x) = 1, and for n > 0, bn=2c bn=2c n X (n − k − 1)! X n − k T (x) = (−1)k (2x)n−2k;U (x) = (−1)k (2x)n−2k n 2 k!(n − 2k)! n k k=0 k=0 Moreover, for all n ≥ 0, sin((n + 1)#) T (cos(#)) = cos(n#);U (cos(#)) = n n sin # The Chebyshev polynomials are a particular case of the Gegenbauer polynomials n (α) (1) Tn(x) = Cn (x) (for n ≥ 1);Un(x) = Cn (x) 2α α=0 Orthogonality 1 1 p dx π 2 π Tm(x)Tn(x) p = (1 + δm;0)δm;n; Um(x)Un(x) 1 − x dx = δm;n 2 ˆ−1 1 − x 2 ˆ−1 2 The Chebyshev polynomials fTn(x)g form an orthogonal basis of H = L2((−1; 1); (1 − x2)−1=2dx), The Chebyshev polynomials fUn(x)g form an orthogonal basis of H = L2((−1; 1); (1 − x2)1=2dx).
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