Related papers: A note on Dickson polynomials of the third kind an…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
In this paper, we have considered second order non-homogeneous linear differential equations having entire coefficients. We have established conditions ensuring non-existence of finite order solution of such type of differential equations.
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the…
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type,…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
We present a general theory for studying the difference analogues of special functions of hypergeometric type on the linear-type lattices, i.e., the solutions of the second order linear difference equation of hypergeometric type on a…
We introduce a new class of polynomials $\{P_{n}\}$, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with $n+1$ unit masses. We study algebraic,…
We study second order and third order linear differential equations with analytic coefficients under the viewpoint of finding formal solutions and studying their convergence. We address some untouched aspects of Frobenius methods for second…
Expressions for the derivatives of the Legendre polynomials of the first kind with respect to the order of these polynomials are given. An explicit form for the fourth derivative is presented.
We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind $ \frac{d}{dt}\tilde{u}(t) = \tilde{f}(t) \tilde{u}(t)$, $\tilde{u}(-1)=1$, with $\tilde{f}(t)$ an analytic…
Clifford-Legendre and Clifford-Gegenbauer polynomials are eigenfunctions of certain differential operators acting on functions defined on $m$-dimensional euclidean space ${\mathbb R}^m$ and taking values in the associated Clifford algebra…
We express a certain complex-valued solution of Legendre's differential equation as the product of an oscillatory exponential function and an integral involving only nonoscillatory elementary functions. By calculating the logarithmic…
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively.…
This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously…
We study the Dickson polynomials of the (k+1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional L_{k}. We find an integral…
In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…
It is well-known that separation of variables in 2nd order partial differential equations (PDEs) for physical problems with spherical symmetry usually leads to Cauchy's differential equation for the radial coordinate and Legendre's…
For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…
We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in algebra of new generalized functions. Then the solution of such equation will be a new generalized function. In the…
Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for…