Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R3 . It is shown that the unknot with maximal Thurston–Bennequin invariant of −1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2–component Legendrian links w...

Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.

The derivatives of the Gaussian function, G(x) = 1 √ 2π e−x 2/2, produce the Hermite polynomials by the relation, (−1)mG(m)(x) = Hm(x)G(x), m = 0, 1, . . . , where Hm(x) are Hermite polynomials of degree m. The orthonormal property of the Hermite polynomials, 1 m! ∫∞ −∞Hm(x)Hn(x)G(x)dx = δmn, can be considered as a biorthogonal relation between the derivatives of the Gaussian, {(−1)nG(n) : n = ...

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the Eulerian polynomials and the shifted Eulerian polynomials are moment sequences for a simple family of orthogonal polynomials. The coefficient arrays of these families of orthogonal polynomials are shown to be exponential Riordan arrays. Using the theory of orthogonal polynomials we are then able t...

introduction 3d reconstruction of an object from its 2d cross-sections (slices) has many applications in different fields of sciences such as medical physics and biomedical engineering. in order to perform 3d reconstruction, at first, desired boundaries at each slice are detected and then using a correspondence between points of successive slices surface of desired object is reconstructed. mate...

The relation between the roots of a polynomial P (z) and its coefficients was one of the driving forces in the development of Algebra. The desire to obtain a closed-form expression for the roots of P (z) = 0 ended with a negative solution at the hands of N. Abel and E. Galois at the beginning of the 19 century: a generic polynomial equation of degree 5 or more cannot be solved by radicals. The ...

Gour C. Karmakar, Laurence S. Dooley, and John Arkinstall Bezier curves (BC) have become fundamental tools in many challenging and varied applications, ranging from computer aided geometric design to generic object shape descriptors. A major limitation of the classical Bezier curve however, is that only global information about its control points (CP) is considered, so there can often be a larg...