Some Properties Involving q-Hermite Polynomials Arising from Differential Equations and Location of Their Zeros

Abstract

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers.

Orthogonality Property of the Discreteq-Hermite Matrix Polynomials

“…Theorem 2. Assume that n, m ∈ N 0 and A is a square complex matrix satisfying the conditions (3) and (4). en, we have 〈H n (x, A; q), H m (x, A; q)〉 � 0, n ≠ m.…”

“…Summarization of the results obtained in this section can be stated in the following theorem: □ Theorem 4. Assume that n ∈ N 0 and A be a square complex matrix satisfying the conditions (3) and (4). en, the discrete q-Hermite matrix polynomial is an orthogonal polynomial with respect to the inner product 〈•, •〉 defined in (47).…”

“…e beginning of q-polynomials is related to the work of Rogers and Ramanujan of the late 19th and early 20th century, see [1,2]. is area of mathematics has grown into an interesting level due to the connection work between Rogers-Ramanujan identities and certain families of orthogonal q-polynomials such as q-Hermite and q-Ultraspherical which have been introduced by Andrews, Askey, Ismail, and Bressoud [3,4]. e classical orthogonal polynomials have many applications in the theory of linear control systems, in robotics and computer aided geometric design [5,6].…”

On second order q-difference equations for high-order Sobolev-type q-Hermite orthogonal polynomials

“…where q stands for their unique parameter, for which we assume that 0 < q < 1, which means that they belong to the class of orthogonal polynomial solutions of certain second order q-difference equations, known in the literature as the Hahn class (see [19], [25]). A recent study of their zeros and other interesting properties has been carried out in [33]. This sequence of polynomials lie at the bottom of the Askey-scheme of hypergeometric orthogonal polynomials, and they are orthogonal with respect to the measure dµ = (qx, −qx; q) ∞ d q x.…”

Preface to the Special Issue “Mathematical Modeling with Differential Equations in Physics, Chemistry, Biology, and Economics”

“…In our Special Issue, there is also an SEIR epidemiological model by Husniah et al [7], where the use of convalescent plasma is supposed to reduce the diffusion of a disease such as COVID-19. Some valuable mathematical results are obtained by Ryoo and Kang [8], whose analysis focuses on q-Hermite polynomials arising from certain differential equations.…”