Ground states of a one-dimensional lattice-gas model with an infinite range nonconvex interaction. A numerical study

We consider a lattice-gas model with an infinite range pairwise noncovex interaction. It might be relevant, for example, for adsorption of alkaline elements on W(112) and Mo(112). We study a competition between the effective dipole-dipole and indirect interactions. The resulting ground state phase diagrams are analysed (numerically) in detail. We have found that for some model parameters the phase diagrams contain a region dominated by several phases only with periods up to nine lattice constants. The remaining phase diagrams reveal a complex structure of usually long periodic phases. We also discuss a possible role of surace states in phase transitions.Comment: 16 pages, 5 Postscript figures; Physical Review B15 (15 August 1996), in pres

Acceleration of generalized hypergeometric functions through precise remainder asymptotics

We express the asymptotics of the remainders of the partial sums {s_n} of the generalized hypergeometric function q+1_F_q through an inverse power series z^n n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k} may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z=1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added several references, added comparison to other methods, and added discussion of recursion stabilit

On the differential-difference properties of the extended Jacobi polynomials

We discuss differential-difference properties of the extended Jacobi polynomials $$P n ( x ) = p + 2 F q ( − n , n + λ , a p ; b q ; x ) ( n = 0 , 1 , … ) . {P_n}(x){ = _{p + 2}}{F_q}( - n,n + \lambda ,{a_p};{b_q};x)\quad (n = 0,1, \ldots ).$$ The point of departure is a corrected and reformulated version of a differential-difference equation satisfied by the polynomials P n ( x ) {P_n}(x) , which was derived by Wimp (Math. Comp., v. 29, 1975, pp. 577-581).</p

Properties of the polynomials associated with the Jacobi polynomials

Power forms and Jacobi polynomial forms are found for the polynomials W n ( α , β ) W_n^{(\alpha ,\beta )} associated with Jacobi polynomials. Also, some differential-difference equations and evaluations of certain integrals involving W n ( α , β ) W_n^{(\alpha ,\beta )} are given.</p