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 123 ), = = . x } . ( ) = ) α } α = 0 x ) R ( α ), and α,β ( n ( n F x β ( = 0 one R H { { P ) = ) one ob- β x ); this pro- ( x ( F ); this produce α, β . 1 and ) = x L } ( x ) . Particular cases: − ( B } ) F α,β Example n ( = = 1 one obtains the = 1 and α,β ) = L ) = ( n = 0. { x . , Legrendre polynomials x ( α } P α α ( 2 1 β ) { F F ± ); this produce the Hermite = α,β ( x n = α ( , Chebyshev I and II if B β then one obtains the Romanovski polynomials, denoted β for each pair ( tains the Jacobi polynomials, de- noted Gegenbauer polynomials if When Hermite equation, then When Eq.(6) isconsidering the Jacobi equation When Eq.(10) is the Romanovskition equa- considering obtains the Laguerrethen equation, duce the Laguerredenoted polynomials, if H polynomials, denoted one obtains the Besselthen equation, the Bessel polynomials, denoted { (5) (3) (7) ) function of eq.(1) x ( s ) = 0 (6) x ( ) (11) ) = 0 (10) x λF ) = 0 (8) ) = 0 (4) x 1 x ( x − ( ( ) + β x λF ) λF ( λF ) (9) x ′ ) = 0 (2) αcot x β x F − ) + ( ] ) + ( ) + x − x x ( ( x (1 + ( ′ 2 ( λF αx ′ α ′ exp F ) − αx F 1 ) F exp e x + 2) ] − − α ) + α β e β β β − x x x ) ( + 2 ′ + + 1) + x ) = x β ) = ) = α βx x ( x x ( + αxF ( ( ) = (1 ) w 2 − + 2) x w ( αx α ) − α α,β ) + (2 ( ) − x − ) = (1 + ( x w α,β x ( ( ′′ β ( ′′ ) w ) + [( F ) + ( F 2 x x ) ( α,β ( ) + [ ( x ′′ Canonical form and weight function ′′ x w ( F 2 ′′ xF x F (1 + ) 2 x − Table 1: Polynomials obtained depending on the (1 ) x ( root s First roots degree with one with two with two Constant double real different real complex roots Second degree: Second degree: Second degree: 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 . . . . an (−n2 − 2λn)an
For simplicity, we take the matrix 4x4 of the Gegenbauer differential oper- ator:
d2 d (1 − x2) − (2λ + 1)x → dx2 dx
0 0 2 0 0 −(2λ + 1) 0 6 (31) 0 0 −4(λ + 1) 0 0 0 0 −3(2λ + 3)
The eigenvalues of a matriz M are the values that satisfy the equation Det(M − λ′I) = 0. However since the matrix (31) is a triangular matrix, ′ the eigenvalues λi of this matrix are the elements of the diagonal, namely: ′ ′ ′ ′ λ1 = 0, λ2 = −(2λ + 1), λ3 = −4(λ + 1), λ4 = −3(2λ + 3). The corresponding ′ eigenvectors are the solutions of the equation (M − λiI) · v = 0, where the T eigenvector v = [a0, a1, a2, a3] .
′ 0 − λi 0 2 0 ′ 0 −(2λ + 1) − λi 0 6 ′ 0 0 −4(λ + 1) − λi 0 ′ 0 0 0 −3(2λ + 3) − λi 128 M. Ram´ırez
a0 0 a 0 · 1 = (32) a 0 2 a3 0
′ Substituing the eigenvalue λ1 in the equation (32) we obtain the eigenvector v1:
1 0 v = (33) 1 0 0 The elements of this eigenvector corresponds to the first Gegenbauer Poly- λ nomial, C0 (x) = 1.
′ Substituing the eigenvalue λ2 in the equation (32) we obtain the eigenvector v2:
0 2λ v = (34) 2 0 0 The elements of this eigenvector corresponds to the second Gegenabuer λ Polynomial, C1 (x) = 2λx.
′ Substituing the eigenvalue λ3 in the equation (32) we obtain the eigenvector v3:
−λ 2λ(λ + 1) v = (35) 3 0 0 The elements of this eigenvector corresponds to the third Gegenabuer Poly- λ 2 nomial, C2 (x) = −λ + 2λ(1 + λ)x .
′ Substituing the eigenvalue λ4 in the equation (32) we obtain the eigenvector v4: SIMPLE APPROACH TO GEGENBAUER POLYNOMIALS 129
0 −2λ(1 + λ) v = (36) 3 0 4 3 λ(1 + λ)(2 + λ) The elements of this eigenvector corresponds to the fourth Gegenabuer Poly- λ 4 3 nomial, C3 (x) = −2λ(1 + λ)x + 3 λ(1 + λ)(2 + λ)x .
4. Conclusion
Gegenbauer polynomials are obtained using basic linear algebra concepts such the eigenvalue and eigenvector of a matrix. Once the corresponding matrix of the Gegenbauer differential operator is obtained, the eigenvalues of this matrix are found and the elements of its eigenvectors correspond to the coefficients of Gegenbauer Polynomials.
5. Acknowledgment
The inspiration and support of Prof. Vicente Aboites along this work is with gratitude sincerely acknowledged. The author is fully responsible for the writing of this article and its contents.
References
[1] V. Aboites, Hermite polynomials through linear algebra, International Journal of Pure and Applied Mathematics 114 (2017), 401-406. [2] A. Raposo, H. J. Weber, D. Alvarez-Castillo, M. Kirchbach, Romanovski polynomials in selected physics problems, Open Physics 5 (2007), 253-284. [3] R. E. Attar, Special functions and orthogonal polynomials, Lulu Press, USA(2007). [4] V. Aboites, Laguerre polynomials and linear algebra, Memorias Sociedad Matem´atica Mexicana 52 (2017), 3-13. 130