Distributional solutions of the hypergeometric differential equation

We present the distributional solutions to the hypergeometric differential equation. These solutions are obtained in the form of infinite series of the Dirac Delta functions and its derivatives. We employ these solutions to observe their interesting features. Furthermore, the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials.     

Hyperstability of a functional equation

The aim of this paper is to prove that the parametric fundamental equation of information is hyperstable on its open as well as on its closed domain, assuming that the parameter is negative. As a corollary of the main result, it is also proved that the system of equations that defines the alpha-recursive information measures is stable.     

Distributional method for the D′ Alembert equation

We consider the D′ Alembert equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation.Received: 22 July 2004; revised: 12 October 2004     

Regularization of a three-element functional equation

In this paper we study the three-element functional equation

$(V\Phi )(z) \equiv \Phi (iz) + \Phi ( - iz) + G(z)\Phi \left( {\frac{1}{z}} \right) = g(z), z \in R,$

, subject to

$R: = \{ z:\left| z \right| < 1, \left| {\arg z} \right| < \frac{\pi }{4}\} .$

We assume that the coefficients G(z) and g(z) are holomorphic in R and their boundary values G ⁺(t) and g ⁺(t) belong to H(Γ), G(t)G(t ^?1) = 1. We seek for solutions Φ(z) in the class of functions holomorphic outside of \(\bar R\) such that they vanish at infinity and their boundary values Φ^?(t) also belong to H(Γ). Using the method of equivalent regularization, we reduce the problem to the 2nd kind integral Fredholm equation.

     

An analog of the Hardy-Littlewood equation

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 58–67, September, 1989.     

Stability of a quartic functional equation

In this paper, the authors investigate the general solution and generalized Hyers–Ulam stability of the n-dimensional quartic functional equation of the form

$$\begin{aligned} f\left( \sum _{i=1}^{n}x_i\right)&= \sum _{1 \le i<j< k< l\le n} f\left( x_i+x_j+x_k+x_l\right) +\left( -n+4\right) \nonumber \\ {}&\sum _{1 \le i< j< k \le n} f\left( x_i+x_j+x_k\right) +\left( \frac{n^2-7n+12}{2}\right) \sum _{ \begin{array}{c} 1=i;\\ i\ne j \end{array}}^{n} f\left( x_i+x_j\right) \nonumber \\&- \sum _{i=1}^{n} f\left( 2x_i\right) + \left( \frac{-n^3+9n^2-26n+120}{6}\right) \ \ \sum _{i=1}^{n}\left( \frac{f(x_i)+f(-x_i)}{2}\right) \end{aligned}$$

where n is a positive integer with \({\mathbb {N}}- \{0,1,2,3,4\}\). The stability of this quartic functional equation is introduced in Banach space using direct and fixed point methods.

     

On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.     

About solutions of a functional equation

Aequationes mathematicae - We consider the functional equation $$f[F(x,y)] = H[g(x),{\text{ }}h(y)]$$ , whereF andH satisfy certain global or local solvability conditions and prove that topological...     

Distributional equation for Laguerre-Hahn functionals on the unit circle

Let u be a Hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. We establish the equivalence between a Riccati differential equation with polynomial coefficients for F, zAF^′=BF²+CF+D, and a distributional equation for u, , where L is the Lebesgue functional, and the polynomials are defined in terms of the polynomials A,B,C,D.     

Distributional solutions to a generalized wave equation for gravity waves on deep water

The equations governing the linearized small amplitude approximation for gravity waves on deep water can be reformulated by the introduction of a cross-surface differential operator, H, which acts like a square-root of the two-dimensional Laplacian. This yields a single scalar equation for the amplitude of the wave-like motion off a horizontal static surface resulting in a mixed initial and boundary value problem for the wave operator, ∂_tt + c²H. The pressure impulse response for an unperturbed static fluid will be calculated via a formal eigenfunction expansion and it will be shown that this yields a distributional solution. Then, the mixed problem will be generalized to allow for distributional data where the initial data is injected into the non-homogeneous term. By employing eigenfunction representations for distributions with compact support it will be shown that a formal eigenfunction expansion also yields a distributional solution to this generalized mixed problem.     

On a general bilinear functional equation

Aequationes mathematicae - Let X,&nbsp;Y be linear spaces over a field $${\mathbb {K}}$$ . Assume that $$f :X^2\rightarrow Y$$ satisfies the general linear equation with respect to the first...