On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are e...

A formula for evaluation of the distribution of a linear combination of independent inverted gamma random variables by one-dimensional numerical integration is presented. The formula is direct application of the inversion formula given by Gil–Pelaez [4]. This method is applied to computation of the generalized p-values used for exact significance testing and interval estimation of the parameter...

Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Perio...

Introduction: Gamma cameras are proposed for online range verification and treatment monitoring in proton therapy.  An Analytical formula was derived and validated for sensitivity of a slit collimator based on the photon fluence concept. Methods: Fluence formulation was generalized for photons distribution function and solved for high-energy point sources. The...

Let K be a smooth immersed curve in the plane. Fabricius-Bjerre [2] found the following relation among the double tangent lines, crossings, and inflections points for a generic K : T1 − T2 = C + (1/2)I where T1 and T2 are the number of exterior and interior double tangent lines of K , C is the number of crossings, and I is the number of inflection points. Here “generic” means roughly that the i...

In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx) Fractional B-splines are a natural extension of classical B-splines. In this short paper, we show their relations to fractional divided differences and fractional difference operators , and present a generalized Hermite-Genochi formula. This formula then allows the definition of a larger class of fractional B-splines.

We present an asymptotic formula for the number of line segments connecting q + 1 points of an n × n square grid, and a sharper formula, assuming the Riemann hypothesis. We also present asymptotic formulas for the number of lines through at least q points and, respectively, through exactly q points of the grid. The well-known case q = 2 is so generalized.

We determine the relations of power product of physical quantities (PQs) by the generalized relation. Assuming the relations with two, three and four PQs to generalized relation separately, we find the corresponding general formulas. By ordering the exponent of physical constant in relations being zero or fit number, we gain many famous equations without factors such as mass-energy relation, fo...