Unification of Bessel functions of different orders

We investigate the internal space of Bessel functions which is associated to the group Z of positive and negative integers defining their orders. As a result we propose and prove a new unifying formula (to be added to the huge literature on Bessel functions) generating Bessel functions of real orders out of integer order one's. The unifying formula is expected to be of great use in applied mathematics. Some applications of the formula are given for illustration.Comment: 8 pages,Late

Mapping Integer Order Neumann Functions To Real Orders

In a recent paper we unified Bessel functions of different orders .Here we extend the unification to other linairely independant solutions to Bessel equation, Neumann's and Hankel's functionsComment: 6 pages, Late

A deformation of Hermite polynomials

We propose and study the properties of a set of polynomials $M_{n\alpha, H\ }^{s}(z)$,$C_{n\alpha, H}^{s}(z)$$W_{n\alpha, H}^{s}(z)$with$n,s\in N$$;\alpha =\pm 1;$and where$H$stands for Hermite ; the ''root '' polynomial >.These polynomials are obtained from a deformation of Hermite polynomials$H_{n}(z).$The structure underlying the deformation seems quite general and not only restricted to Hermite polynomials.Comment: 16 pages, Late

The ML(z);CL(z);WL(z)M_{L}(z);C_{L}(z);W_{L}(z) associated Laguerre Polynomials

In a previous paper we deformed Hermite polynomials to three associated polynomials .Here we apply the same deformation to Laguerre polynomials .Comment: Latex 2e, 12 page

Cohomological Quantum Mechanics And Calculability Of Observables

We reconsider quantum mechanical systems based on the classical action being the period of a one form over a cycle and elucidate three main points. First we show that the prepotenial V is no longer completely arbitrary but obeys a consistency integral equation. That is the one form dV defines the same period as the classical action. We then apply this to the case of the punctured plane for which the prepotential is of the form $V= \alpha \theta + \Phi ( \theta )$. The function $\Phi$ is any but a periodic function of the polar angle. For the topological information to be preserved, we further require that $\Phi$ be even. Second we point out the existence of a hidden scale which comes from the regularization of the infrared behaviour of the solutions. This will then be used to eliminate certain invariants preselected on dimensional counting grounds. Then provided we discard nonperiodic solutions as being non physical we compute the expectation values of the BRST- exact observables with the general form of the prepotential using only the orthonormality of the solutions (periodic). Third we give topological interpretations of the invariants in terms of the topological invariants wich live naturally on the punctured plane as the winding number and the fundamental group of homotopy,but this requires a prior twisting of the homotopy structure.Comment: 12 pages,Late

The Trace Formula of the Spinoriel Amplitude

We re express the fermion's probability amplitude as a trace over spinor indices, which formulation surprisingly does not exist in literature. This formulation puts the probabilty amplitude and the the probabilty(squared amplitude) of a given process on equal footing at the compuational level and this is our principal motivation to write the present paper. We test the power of the trace formula in three applications: Calculation of the charge-current of fermions by using symbolic programs, which current so far was only computable by hand, analytic compuation of the quark dipole magnetic moment, rendered less cumbersome, and finally Fiertz rearrangement identities now made more transparent.Comment: Work presented at: HEP 2009 16-22 July 2009, Krakow, Poland AND Dspin-09,Dubna,Russia,Septembre 1-5, 200

Witten deformed exterior derivative and Bessel functions

In a recent paper we investigated the internal space of Bessel functions associated with their orders. We found a formula (new) unifying Bessel functions of integer and of real orders. In this paper we study the deformed exterior derivative system $H=d_{\lambda}$ on the puctured plane as a tentative to understand the origin of the formula and find that indeed similar formula occurs. This is no coincidence as we will demonstrate that generating functions of integer order Bessel functions and of real orders are respectively eigenstates of the usual exterior derivative and its deformation. As a direct consequence we rediscover the unifying formula and learn that the system linear in $d_{\lambda}$ is related to Bessel theory much as the system quadratic in ($d_{\lambda}+d_{\lambda}^{*}$) is related to Morse theory.Comment: 8 pages late

Magnetic moment versus tensor charge

We express the baryon magnetic moments in terms of the baryon tensor charges, considering the quarks as relativistic interacting objects. Once tensor charges get measured accurately, the formula for the baryon magnetic moment will serve to extract precise information on the quark anomalous magnetic moment, the quark effective mass and the ratio of the quark constituent mass to the quark effective mass. The analogous formula for the baryon electric dipole moment is of no great use as it gets eventually sizable contributions from various CP- violating sources not necessary associated to the quark electric dipole moment.Comment: 15 pages pdf forma

Cohomology and Bessel functions Theory

By studying cohomological quantum mechanics on the punctured plane,we were led to identify (reduced) Bessel functions with homotopic loops living on the plane.This identification led us to correspondence rules between exponentials and Bessel functions.The use of these rules makes us retrieve known but also new formulas in Bessel functions theory.Comment: 11 pages .Latex2e.Talk given at Group24 Paris 15-20 July 200

Twisted Homotopy: A Group Theoretic Approach

After summarising the physical approach leading to twisted homotopy and after developing the cohomological approach further with respect to our previous work we propose a third alternative approach to twisted homotopy based on group theoretic considerations. In this approach the fundamental group $\Pi (m)$ isomorphic to Z which describes homotopic loops on the punctured plane$R^2/(0)$ is enhanced in a special way to the continuous SO(2) group . This is performed by letting the parameter of the group $m \rightarrow \lambda$ while keeping its generator unchanged .It is shown that such non-trivial procedure has the effect of introducing well defined self-interactions among loops which are at the basis of twisted homotopy where the angle $\lambda$ plays the role of the self coupling constant. KEYWORDS: Homotopy, Group Theory, Quantum Mechanics MSC:55Q35; PACS:02.20.Fh ; 03.65.FdComment: 7 pages,Latex,no figure