We observe that the exact connection coefficient relations transforming modal coefficients of one Jacobi Polynomial class to the modal coefficients of certain other classes are sparse. Because of this, when one of the classes corresponds to the Chebyshev case, the Fast Fourier Transform can be used to quickly compute modal coefficients for Jacobi Polynomial expansions of class (α, β) when 2α an...

We study the structure of solutions of the one-dimensional non-linear pseudodifferential equation describing the dynamics of the p-adic open string for the scalar tachyon field p 1 2 ∂ t Φ = Φ. We elicit the role of real zeros of the entire function Φ(z) and the behaviour of solutions Φ(t) in the neighbourhood of these zeros. We point out that discontinuous solutions can appear if p is even. We...

Three novel and significant contributions are made in this paper Firstly, non-recursive formulation of Haar connection coefficients, pioneered by the present authors is presented, which can be computed very efficiently and avoid stack and memory overflows. Secondly, the generalized approach for state analysis of singular bilinear time-invariant (TI) and time-varying (TV) systems is presented; v...

This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition ν, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when ν = (n) ...

The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-ord...

A connection is made between complete homogeneous symmetric polynomials in Jucys-Murphy elements and the unitary Weingarten function from random matrix theory. In particular we show that hr(J1,...,Jn), the complete homogeneous symmetric polynomial of degree r in the JM elements, coincides with the rth term in the asymptotic expansion of the Weingarten function. We use this connection to determi...

By making the connection between four-dimensional lattice Green functions (LGFs) and Picard–Fuchs ordinary differential equations of Calabi–Yau manifolds, we have given explicit forms for the coefficients of the fourdimensional LGFs on the simple-cubic and body-centred cubic lattices, in terms of finite sums of products of binomial coefficients, and have shown that the corresponding four-dimens...