Apéry’s proof [11] of the irrationality of ζ(3) is now over 25 years old, and it is perhaps surprising that his methods have not yielded any significantly new results (although further progress has been made on the irrationality of zeta values [1], [12]). Shortly after the initial proof, Beukers produced two elegant reinterpretations of Apéry’s arguments; the first using iterated integrals and ...

The 12 inverse Jacobian elliptic functions are expanded in power series by using properties of the symmetric elliptic integral of the first kind. Suitable notation allows three series to include all 12 cases, three of which have been given previously. All coefficients are polynomials in the modulus k that are homogeneous variants of Legendre polynomials. The four series in each of three subsets...

In this paper we describe an efficient algorithm for computing the potentials of the form r−λ where λ ≥ 1. This treecode algorithm uses spherical harmonics to compute multipole coefficients that are used to evaluate these potentials. The key idea in this algorithm is the use of Gegenbauer polynomials to represent r−λ in a manner analogous to the use of Legendre polynomials for the expansion of ...

Identification of nonlinear restoring force and dynamic loadings provides critical information for post-event damage diagnosis structures. Due to high complexity individuality structural nonlinearities, it is difficult provide an exact parametric mathematical model in advance describe the behavior a member or substructure under strong practice. Moreover, external loading applied engineering str...

This paper provides the details of Remark 5.4 in the author’s paper “Askey-Wilson polynomials as zonal spherical functions on the SU(2) quantum group”, SIAM J. Math. Anal. 24 (1993), 795–813. In formula (5.9) of the 1993 paper a two-parameter class of Askey-Wilson polynomials was expanded as a finite Fourier series with a product of two 3phi2’s as Fourier coefficients. The proof given there use...

In this paper, we present a characterization of a semi-multiplicative analogue of planar functions over finite fields. When the field is a prime field, these functions are equivalent to a variant of a doubly-periodic Costas array and so we call these functions Costas. We prove an equivalent conjecture of Golomb and Moreno that any Costas polynomial over a prime field is a monomial. Moreover, we...

where aij ∈ K. Additive polynomials over valued fields in positive characteristic play an important role in understanding many algebraic and model theoretic properties of maximal fields of positive characteristic, see [7] for a thorough examination of the issue. A subset S of a valued field (K, v) has the optimal approximation property if for all a ∈ K, the set {v(s − a) : s ∈ S} has a maximal ...

In this paper, we investigate the spectral method on quadrilaterals. We introduce an orthogonal family of functions induced by Legendre polynomials, and establish some results on the corresponding orthogonal approximation. These results play important roles in the spectral method for partial differential equations defined on quadrilaterals. As examples of applications, we provide spectral schem...