On generalized Heun equation with some mathematical properties

We study the analytic solutions of the generalized Heun equation, (α0 + α1 r + α2 r2 + α3 r3) y′′ + (β0 + β1 r + β2 r2) y′ + (ε0 + ε1 r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple procedure based on a recurrence relation is introduced to evaluate these polynomial solutions explicitly. For α0 = 0, α1≠ 0, we prove that the polynomial solutions of the corresponding differential equation are sources of finite sequences of orthogonal polynomials. Several mathematical properties, such as the recurrence relation, Christoffel-Darboux formulas and the norms of these polynomials, are discussed. We shall also show that they exhibit a factorization property that permits the construction of other infinite sequences of orthogonal polynomials

Sextic anharmonic oscillators and orthogonal polynomials

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.Comment: 11 page

Prepositions in MSA and English

Spatial scenes are identical in the world languages. However, cultures may diverge in profiling spatial scenes (Levinson 2003). This paper selects for study the prepositions in and on in English and their Modern Standard Arabic (MSA) counterparts fi and 3ala, arguing that MSA and English seem to diverge in the spatial configurations and meanings of these prepositions. The sub-schemas of CONTAINMENT (in-ness) in MSA are found to partially overlap with those of English, with the other sub-schemas being taken care of by SUPPORT (on-ness) and PUNCTUALITY (point-ness). Such differences classify MSA more as a CONTAINMENT-based language than English, which seems to prefer SUPPORT and PUNCTUALITY. However, English and MSA seem to converge in their metaphoric conceptualizations of states owing to conceptual embodiment (Lakoff 1987). The article discusses the implications of such findings for spatial cognition and cultural cognition and EFL/ESL writing and translating