Related papers: \pi and other formulae implied by hypergeometric s…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a…
A certain special function of the generalized hypergeometric variety is shown to fulfill a host of useful noncommutative identities.
We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem,…
Decomposition formulas associated with the Lauricella multivariable hypergeometric functions were known, however, due to the recurrence of those formulas, additional difficulties may arise in the applications. Further study of the…
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
It has been shown earlier that the solubility of the Legendre and the associated Legendre equations can be understood as a consequence of an underlying supersymmetry and shape invariance. We have extended this result to the hypergeometric…
Classical linear wave superposition produces the appearance of interference. This observation can be interpreted in two equivalent ways: one can assume that interference is an illusion because input components remain unperturbed, or that…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
Using some new logarithmic formal calculus, we construct a well known vertex algebra, obtaining the Jacobi identity directly, in an essentially self-contained treatment.
In this paper, we introduce formal sine functions whose coefficients are elements of a generalized harmonic algebra and investigate their properties corresponding to the classical addition formula and Pythagorean theorem. By taking their…
In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended…
This paper develops second variational formulas and index forms in the context of Hermitian geometry. Building upon these analytical foundations, we establish results analogous to classical theorems in Riemannian geometry, including Myers'…
By systematically applying ten inequivalent two-part relations between hypergeometric sums 3F2(1) to the published database of all such sums, 66 new sums are obtained. Many results extracted from the literature are shown to be special cases…
A family of original formulae for computing number PI and its proof are presented. An algorithm is proposed to validate the results of this new algorithm.
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study…
I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
Work in progress concerning alternative formalizations of arithmetic.