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Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Zagorodniuk, S. (2001)

Serdica Mathematical Journal

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2...

Distribution of zeros and shared values of difference operators

Jilong Zhang, Zongsheng Gao, Sheng Li (2011)

Annales Polonici Mathematici

We investigate the distribution of zeros and shared values of the difference operator on meromorphic functions. In particular, we show that if f is a transcendental meromorphic function of finite order with a small number of poles, c is a non-zero complex constant such that Δ c k f 0 for n ≥ 2, and a is a small function with respect to f, then f Δ c k f equals a (≠ 0,∞) at infinitely many points. Uniqueness of difference polynomials with the same 1-points or fixed points is also proved.

Équations aux différences associées à des groupes, fonctions représentatives.

Nicolas Marteau (2004)

Annales de l’institut Fourier

Inspiré par un travail de J.-P. Bézivin et F. Gramain sur les systèmes d’équations aux différences, on caractérise les sous-groupes H d’un groupe de Lie réel (resp. complexe) G , pour lesquels toute fonction f : G continue (resp. entière) telle que l’ensemble des H -translatées engendrent un -espace vectoriel de dimension finie, engendrent aussi un -espace vectoriel de dimension finie par G - translation. On fait le lien avec les systèmes d’équations aux différences à coefficients constants.

Existence and positivity of solutions for a nonlinear periodic differential equation

Ernest Yankson (2012)

Archivum Mathematicum

We study the existence and positivity of solutions of a highly nonlinear periodic differential equation. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ a modification of Krasnoselskii’s fixed point theorem introduced by T. A. Burton ([4], Theorem 3) to show the existence and positivity of solutions of the equation.

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