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Simple Approach to Gegenbauer Polynomials

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Simple Approach to Gegenbauer Polynomials International Journal of Pure and Applied Mathematics Volume 119 No. 1 2018, 121-129 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v119i1.10 ijpam.eu SIMPLE APPROACH TO GEGENBAUER POLYNOMIALS Miguel Ram´ırez Divisi´on de Ciencias e Ingenier´ıas Universidad de Guanajuato Loma del Bosque 103, Col. Campestre, 37150 Le´on, MEXICO´ Abstract: Gegenbauer polynomials are obtained through well known linear algebra methods based on Sturm-Liouville theory. A matrix corresponding to the Gegenbauer differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained corresponds to the Gegenbauer polynomials. AMS Subject Classification: 97x01, 42C05, 34L15, 33C45 Key Words: Gegenbauer, special functions, Gegenbauer polynomials 1. Introduction Gegenbauer polynomials are solutions of an ordinary differential equation (ODE) which is an hypergeometric equation. In general an hypergeometric equation may be writen as: ′′ ′ s(x)F (x) + t(x)F (x) + λF (x) = 0 (1) where F (x) is a real function of a real variable F : U −→ R, where U ⊂ R is an open subset of the real line, and λ ∈ R a corresponding eigenvalue, and the functions s(x) and t(x) are real plynomials of at most second order and first order, respectively. The Sturm-Liouville Theory is covered in most advanced courses. In this context an eigenvalue equation sometimes takes the more general self-adjoint form: L u(x) + λw(x)u(x) = 0, where L is a differential operator; L u(x) = d du(x) dx hp(x) dx i + q(x)u(x), λ an eigenvalue, and w(x) is known as a weight Received: October 22, 2017 c 2018 Academic Publications, Ltd. Revised: March 11, 2018 url: www.acadpubl.eu Published: June 10, 2018 122 M. Ram´ırez or density function. The analysis of this equation and its solutions is called Sturm-Liouville theory. Specific forms of p(x), q(x), λ and w(x), are given for Legendre, Laguerre, Hermite and other well-known equations in the given references. There, it is also shown the close analogy of this theory with linear algebra concepts. For example, functions here take the role of vectors there, and linear operators here take that of matrices there. Finally, the diagonalization of a real symmetric matrix corresponds to the solution of an ordinary differential equation, defined by a self-adjoint operator L , in terms of its eigenfunctions which are the ”continuous” analog of the eigenvectors [4]. There are different cases obtained depending on the kind of the s(x) function in eq.(1). When s(x) is a constant, eq.(1) takes the form F ′′(x) − 2αxF ′(x) + λF (x) = 0, and if α = 1 one obtains the Hermite Polynomials. When s(x) is a polynomial of first degree, eq.(1) takes the form xF ′′(x)+(−αx+β +1)F ′(x)+ λF (x) = 0, and when α = 1 and β = 0 one obtains the Laguerre Polynomials. There are three different cases when s(x) is a polynomial of second degree. When the second degree polynomial has two different real roots, eq.(1) takes the form (1 − x2)F ′′(x) + [β − α − (α + β + 2)x] F ′(x) + λF (x) = 0 and that is the Jacobi equation, for different values of α and β one obtains particular cases of polynomials; Gegenbauer polynomials if α = β; Chebyshev I and II 1 if α = β = ± 2 ; Legrendre polynomials if α = β = 0. When the second degree polynomial has one double real root, eq.(1) takes the form x2F ′′(x) + [(α + 2)x + β] F ′(x) + λF (x) = 0, and when α = −1 and β = 0 one obtains the Bessel Polynomials. At last, when the second degree polynomial has two complex roots, eq.(1) takes the form (1+x)2F ′′(x)+(2βx+α)F ′(x)+λF (x) = 0, and that is the Romanovski equation. These results are sumarized in Table 1 [2]. 2. Gegenbauer Polynomials λ The Gegenbauer Polynomials, denoted by Cn (x), are particular solutions of the Gegenbauer differential equation and at the same time are a special case of Jacobi polynomials. The Gegenbauer differential equation is given by: ′′ ′ (1 − x2)y − (2λ − 1)xy + n(n + 2λ)y = 0 (12) Gegenbauer Polynomials are called sometimes Ultraspherical Polynomials [3]. The first few polynomials are given by: λ C0 (x) = 1 (13) SIMPLE APPROACH TO GEGENBAUER POLYNOMIALS s(x) Canonical form and weight function Example Constant When α = 1 one obtains the ′′ ′ F (x) − 2αxF (x) + λF (x) = 0 (2) Hermite equation, then F (x) = −αx2 H(x); this produce the Hermite w(x) = e (3) (α) polynomials, denoted {Hn }. First When α = 1 and β = 0 one ′′ ′ degree xF (x) + (−αx + β + 1)F (x) + λF (x) = 0 (4) obtains the Laguerre equation, − then F (x) = L(x); this pro- w(x) = xβe αx (5) duce the Laguerre polynomials, (α,β) denoted {Ln }. Second degree: Eq.(6) is the Jacobi equation with two considering F (x) = P (x), and 2 ′′ ′ different real (1 − x )F (x) + [β − α − (α + β + 2)x] F (x) + λF (x) = 0 (6) for each pair (α, β) one ob- roots (α,β) α β tains the Jacobi polynomials, de- w (x) = (1 − x) (1 + x) (7) (α,β) noted {Pn }. Particular cases: Gegenbauer polynomials if α = β, Chebyshev I and II if α = 1 β = ± 2 , Legrendre polynomials if α = β = 0. Second degree: When α = −1 and β = 0 ′′ ′ with one x2F (x) + [(α + 2)x + β] F (x) + λF (x) = 0 (8) one obtains the Bessel equation, double real β then F (x) = B(x); this produce root w(α,β)(x) = xαexp(− ) (9) the Bessel polynomials, denoted x (α,β) {Bn }. Second degree: Eq.(10) is the Romanovski equa- ′′ ′ with two (1 + x)2F (x) + (2βx + α)F (x) + λF (x) = 0 (10) tion considering F (x) = R(x), complex roots (α,β) 2 β−1 −1 then one obtains the Romanovski w (x) = (1 + x ) exp(−αcot x) (11) (α,β) polynomials, denoted {Rn }. Table 1: Polynomials obtained depending on the s(x) function of eq.(1) 123 124 M. Ram´ırez λ C1 (x) = 2λx (14) λ 2 C2 (x) = −λ + 2λ(1 + λ)x (15) λ 4 C (x) = −2λ(1 + λ)x + λ(1 + λ)(2 + λ)x3 (16) 3 3 Note that if λ = 1/2 the equation (12) reduces to the Legendre equation, and the Gegenbauer Polynomials reduce to the Legendre Polynomials. And, if λ = 1 the equation (12) reduces to the Chebyshev differential equation, and the Gegenbauer Polynomials reduce to the Chebyshev Polynomials of the second kind. In the next section, the Gegenbauer polynomials are obtained not by solving the Gegenbauer differential equation (12), but using a linear algebra method as shown in reference [1] and [4]. 3. Gegenbauer polynomials through matrix algebra Representing the general algebraic polynomial of degree n, as: 2 3 n a0 + a1x + a2x + a3x + ... + anx (17) with a0, a1, a2, . , an ∈ R, by the vector: a0 a1 a 2 An = a (18) 3 . . an Also the first derivative of (17) can be represented as a vector, considering the polynomial: d 2 3 n a + a x + a x + a x + ... + anx = dx 0 1 2 3 2 n−1 a1 + 2a2x + 3a3x + ... + nanx (19) SIMPLE APPROACH TO GEGENBAUER POLYNOMIALS 125 a1 2a2 3a dAn 3 = . (20) dx . nan 0 Now taking the second derivative of (17): 2 d 2 3 n a + a x + a x + a x + ... + anx = dx2 0 1 2 3 2 n−2 2a2 + 6a3x + 12a4x + ... + n(n − 1)anx (21) and, 2a2 6a3 2 . d An . 2 = (22) dx n(n − 1)an 0 0 Using (20), equation (19) may be written as: 0 1 0 0 ··· 0 a0 a1 0 0 2 0 ··· 0 a1 2a2 0 0 0 3 ··· 0 a 3a 2 3 . a = . (23) . .. 3 . . 0 0 0 0 ··· n . nan 0 0 0 0 ··· 0 an 0 Therefore the first derivative operator of An may be written as: 0 1 0 0 ··· 0 0 0 2 0 ··· 0 0 0 0 3 ··· 0 d → . (24) dx . .. 0 0 0 0 ··· n 0 0 0 0 ··· 0 126 M. Ram´ırez Doing the same for equation (21): 0 0 2 0 ··· 0 a0 2a2 0 0 0 6 ··· 0 a1 6a3 . a . . .. 2 . a = (25) 0 0 0 0 ··· n(n − 1) 3 n(n − 1)an . 0 0 0 0 ··· 0 . 0 0 0 0 0 ··· 0 an 0 Therefore the second derivative operator of An may be written as: 0 0 2 0 ··· 0 0 0 0 6 ··· 0 2 . d . .. 2 → (26) dx 0 0 0 0 ··· n(n − 1) 0 0 0 0 ··· 0 0 0 0 0 ··· 0 The Gegenbauer differential operator is given by: d2 d (1 − x2) − (2λ + 1)x (27) dx2 dx substituing (19) and (21) into (27): 2 2 n−2 (1 − x )[2a2 + 6a3x + 12a4x + ... + n(n − 1)anx ] + 2 n−1 −(2λ + 1)x[a1 + 2a2x + 3a3x + ... + nanx ] (28) 2 = 2a2 + [6a3 − (2λ + 1)a1]x + [12a4 − 4(λ + 1)a2]x + 3 2 n +[20a5 − 3(2λ + 3)a3]x + ··· + (−n − 2λn)anx (29) Which may be written as: SIMPLE APPROACH TO GEGENBAUER POLYNOMIALS 127 0 0 2 0 0 0 ··· 0 0 −(2λ + 1) 0 6 0 0 ··· 0 0 0 −4(λ + 1) 0 12 0 ··· 0 0 0 0 −3(2λ + 3) 0 20 ··· 0 . . .. 0 0 0 0 0 0 · · · −n2 − 2λn a0 2a2 a1 6a3 − (2λ + 1)a1 a 12a − 4(λ + 1)a 2 4 2 · a = 20a − 3(2λ + 3)a (30) 3 5 3 .
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