International Journal of Pure and Applied Volume 114 No. 2 2017, 401-406 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v114i2.19 ijpam.eu

HERMITE POLYNOMIALS THROUGH LINEAR ALGEBRA

V. Aboites Center for Research in Optics Loma del Bosque 115 37150 Le´on, MEXICO

Abstract: Hermite Polynomials are obtained thorough linear algebra methods. A matrix corresponding to the Hermite differential operator is found and its eigenvalues are obtained. The elements of the eigenvectors obtained corresponds to the Hermite polynomials.

AMS Subject Classification: 97.01, 34L15, 33B99 Key Words: Hermite, Hermite Polynomials, special functions

1. Introduction

Hermite differential equation and its solutions, i.e. Hermite polynomials, are found in many important physical problems. The study of the quantum har- monic oscillator is an important example [1], another one is the description of Gaussian-Spherical Beams in optics and [2]. Many more applica- tions are found in systems theory in connection with nonlinear operations on Gaussian noise, as well as in , and . Hermite polynomials are studied in most science and engineering mathemat- ics courses, mainly in those courses focused on differential equations or special

Received: April 5, 2017 c 2017 Academic Publications, Ltd. Revised: May 2, 2017 url: www.acadpubl.eu Published: May 8, 2017 402 V. Aboites functions. These polynomials are typically obtained as a result of the solution of Hermite differential equation by or Frobenius method. Usually it is also shown that they can be obtained by a and also by Rodriguez formula for Hermite polynomial. Most courses also include a study of the properties of these polynomials such as: , completeness, recursion relations, special values, asymptotic expansions and relation to other functions such as and hypergeometric functions [3], [4]. There is no doubt that this is a demanding subject that requires a great deal of attention from most students. In this paper Hermite polynomials are ob- tained using basic concepts of linear algebra (which most students are already familiar with) and which contrasts in simplicity with the standard methods as those described in the previously outlined syllabus. In the next section the Hermite differential operator matrix is obtained as well as its eigenvalues and eigenvectors. From the eigenvectors found, the Hermite polynomials follow.

2. Hermite Polynomials

The algebraic polynomial of degree N,

2 3 n a0 + a1x + a2x + a3x + . . . anx , (1) with ao, a1, . . . an ∈ ℜ, is represented by the vector:

a0   a1  a   2  An =  a  . (2)  3   .   .     an  Taking first derivative of the above polynomial (1) one obtains the polyno- mial:

d 2 3 n a + a x + a x +a x + . . . anx ) = dx 0 1 2 3 2 n−1 a1 + 2a2x + 3a3x + . . . nanx , (3)

Which may be written as: HERMITE POLYNOMIALS THROUGH LINEAR ALGEBRA 403

a1   2a2 dAn  3a  =  3  (4) dx  ...     nan     0  Taking the second derivative of polynomial (1) one obtains:

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 (5)

or,

2a2   6a3 d2An  ...  =   . (6) dx2  n(n − 1)an     0     0  Equation (4) may be written as:

0 1 0 0 ... 0 0 a1 a1       0 0 2 0 ... 0 0 a2 2a2  0 0 0 3 ... 0 0   a   3a     3  =  3  (7)  ......   ...   ...         0 0 0 0 ... 0 n   an−   nan     1     0 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 0 2 0 ... 0 0  d  0 0 0 3 ... 0 0  →   (8) dx  ......     0 0 0 0 ... 0 n     0 0 0 0 ... 0 0  In a similar manner, equation (5) may be written as: 404 V. Aboites

a 2a 0 0 2 0 ... 0 0 0 1  a   6a   0 0 0 6 ... 0 0  1 3  a2   12a4   ......         a  =  ...  (9)  0 0 0 0 ... 0 n(n − 1)   3       ...   n(n − 1)   0 0 0 0 ... 0 0         an−   0   0 0 0 0 ... 0 0   1     an   0  Therefore the second derivative operator of An may be written as:

0 0 2 0 ... 0 0  0 0 0 6 ... 0 0  d2  ......  →   (10) dx2  0 0 0 0 ... 0 n(n − 1)     0 0 0 0 ... 0 0     0 0 0 0 ... 0 0  The Hermite differential operator is given by:

d2 d − 2x (11) dx2 dx Which using eq. (8) and (10) may be written as:

2 4 n−2 2a + 6a x + 12a x + 20a x + . . . n(n − 1)an −  2 3 4 5  2 3 n−1 2x a + 2a x + 3a x + 4a x + . . . nan  1 2 3 4  2 3 n−2 = 2a + 6a x + 12a x + 20a x + . . . n(n − 1)an −  2 3 4 5  2 3 n 2a x + 4a x + 6a x + ... 2anx  1 2 3  2 = 2a2 + (6a3 − 2a1)x + (12a4 − 4a2)x 3 + (20a5 + 6a3)x + ... (12) Which may be written as:

a 2a 0 0 2 0 0 ... 0 0 2  a   6a − 2a   0 −2 0 6 0 ... 0  1 3 1  a2   12a4 − 4a2   0 0 −4 0 12 ... 0         a  =  20a − 6a  (13)  0 0 0 −6 0 ... 0   3   5 3     ...   ...   ......         an−   ...   0 0 0 0 0 ... −2n   1     an   −2n  HERMITE POLYNOMIALS THROUGH LINEAR ALGEBRA 405

Therefore, for the sake of simplicity, as a 4×4 matrix the Hermite differential operator is represented by the following matrix:

0 0 2 0 d2 d  0 −2 0 6  − 2x → (14) dx2 dx  0 0 −4 0     0 0 0 −6  The eigenvalues of a matrix M are the values that satisfy the equation (M −λI) = 0. However since Matrix (14) is a triangular matrix, the eigenvalues λi of this matrix are the elements of the diagonal, namely: λ1 = 0, λ2 = −2, λ3 = −4, λ4 = −6 . The corresponding eigenvectors are the solutions T of the equation (M − λiI) · v = 0, where the eigenvector v = [a0, a1, a2, a3] .

0 − λi 0 2 0 a0 0       0 −2 − λi 0 6 a 0 1 = (15)  0 0 −4 − λi 0   a   0     2     0 0 0 −6 − λi   a3   0 

Substituting in equation (15) the first eigenvalue λ1 = 0, one obtains the eigenvector v1:

1  0  v = (16) 1  0     0  The elements of this eigenvector corresponds to the first Hermite polyno- mial, H0(x) = 1. Substituting in equation (15) the second eigenvalue λ2 = −2, one obtain the eigenvector v2:

0  2  v = (17) 2  0     0  The elements of this eigenvector corresponds to the second Hermite poly- nomial, H1(x) = 2x Substituting in equation (15) the third eigenvalue λ3 = −4, one obtain the eigenvector v3: 406 V. Aboites

−2  0  v = (18) 3  4     0  The elements of this eigenvector corresponds to the third Hermite polyno- 2 mial, H2(x) = 4x − 2. Substituting in equation (15) the fourth eigenvalue λ4 = −6, one obtains the eigenvector v4:

0  −12  v = (19) 4  0     8  The elements of this eigenvector corresponds to the fourth Hermite polyno- 3 mial, H3(x) = 8x − 12x. Using a larger matrix, higher order polynomials may be obtained.

3. Conclusion

Hermite polynomials are obtained using basic linear algebra concepts such the eigenvalue and eigenvector of a matrix. Once the corresponding matrix of the Hermite differential operator is obtained, the eigenvalues of this matrix are found and the elements of its eigenvectors correspond to the Hermite Polyno- mials.

References

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables, Dover Publications (2012). [2] P. Frank, R. v. Mises, Die Differential und Integralgleichungen der Mechanik und Physik, Ed. Mary S. Rosenberg, New York, (1943). [3] M.L. Boas, Mathematical Methods in the Physical Sciences, John Wiley & Sons, (2014). [4] G. Arfken, Mathematical Methods for Physicists, Academic Press, (2010).