On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

On Polynomial Solutions of Pell’s Equation

Polynomial Pell’s equation is x 2 − D y 2 = ± 1 , where D is a quadratic polynomial with integer coefficients and the solutions X , Y must be quadratic polynomials with integer coefficients. Let D = a 2 x 2 + a 1 x + a 0 be a polynomial in Z x . In this paper, some quadratic polynomial solutions are given for the equation x 2 − D y 2 = ± 1 which are significant from computational point of view.

The Polynomial Solutions of Quadratic Diophantine Equation X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0

In this study, we consider the number of polynomial solutions of the Pell equation x 2 − p t y 2 = 2 is formulated for a nonsquare polynomial p t using the polynomial solutions of the Pell equation x 2 − p t y 2 = 1 . Moreover, a recurrence relation on the polynomial solutions of the Pell equation x 2 − p t y 2 = 2 . Then, we consider the number of polynomial solutions of Diophantine equation E : X 2 − p t Y 2 + 2 K t X + 2 p t L t Y = 0 . We also obtain some formulas and recurrence relations on the polynomial solution X n , Y n of E .

Asymptotic iteration method for solving Hahn difference equations

Hahn’s difference operator $D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w})$ D q ; w f ( x ) = ( f ( q x + w ) − f ( x ) ) / ( ( q − 1 ) x + w ) , $q\in (0,1)$ q ∈ ( 0 , 1 ) , $w>0$ w > 0 , $x\neq w/(1-q)$ x ≠ w / ( 1 − q ) is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the $(q;w)$ ( q ; w ) -hypergeometric equation.

Laplace Contour Integrals and Linear Differential Equations

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \end{aligned}$$ L [ w ] ( z ) : = w ( n ) + ∑ j = 0 n - 1 ( a j + b j z ) w ( j ) = 0 . This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

Construction Of Three-Variable High-Order Defferential Equation Polynomial Solutions By Combinatorics Method

In this paper, we study how basic systems of polynomial solutions of a differential equation of high order with mixed derivatives of a function of three variables are constructed using combinatorial methods

Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$

For each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ Q [ x ] of smallest degree n that are solutions of the equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ P ( x ) x n + 1 + Q ( x ) ( x + 1 ) n + 1 = 1 . We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.