2,178 research outputs found

    A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

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    In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at x=0x = 0 and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

    Laguerre-type derivatives: Dobinski relations and combinatorial identities

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    We consider properties of the operators D(r,M)=a^r(a^\dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^\dag are boson annihilation and creation operators respectively, satisfying [a,a^\dag]=1. We obtain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.Comment: 14 pages, 1 figur

    Some Useful Collective Properties of Bessel, Marcum Q-Functions and Laguerre Polynomials

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    Special functions have been used widely in many problems of applied sciences. However, there are considerable numbers of problems in which exact solutions could not be achieved because of undefined sums or integrals involving special functions. These handicaps force researchers to seek new properties of special functions. Many problems that could not be solved so far would be solved by means of these efforts. Therefore in this article, we derived some useful properties and interrelations of each others of Bessel functions, Marcum Q-functions and Laguerre polynomials

    Conformal symmetry of the Lange-Neubert evolution equation

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    The Lange-Neubert evolution equation describes the scale dependence of the wave function of a meson built of an infinitely heavy quark and light antiquark at light-like separations, which is the hydrogen atom problem of QCD. It has numerous applications to the studies of B-meson decays. We show that the kernel of this equation can be written in a remarkably compact form, as a logarithm of the generator of special conformal transformation in the light-ray direction. This representation allows one to study solutions of this equation in a very simple and mathematically consistent manner. Generalizing this result, we show that all heavy-light evolution kernels that appear in the renormalization of higher-twist B-meson distribution amplitudes can be written in the same form.Comment: 8 page

    SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND FOR SOME SEQUENCES AND FUNCTIONS: Special values of Bell polynomials of second kind

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    In the paper, the authors concisely review some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, explicitly present closed formulas for those sequences investigated in [F. T. Howard, A special class of Bell polynomials, Math. Comp. 35 (1980), no. 151, 977–989; Available online at https://doi.org/10.2307/2006208], and newly establish some closed formulas for some special values of the Bell polynomials of the second kind

    The Zagier polynomials. Part II: Arithmetic properties of coefficients

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    The modified Bernoulli numbers \begin{equation*} B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 \end{equation*} introduced by D. Zagier in 1998 were recently extended to the polynomial case by replacing BrB_{r} by the Bernoulli polynomials Br(x)B_{r}(x). Arithmetic properties of the coefficients of these polynomials are established. In particular, the 2-adic valuation of the modified Bernoulli numbers is determined. A variety of analytic, umbral, and asymptotic methods is used to analyze these polynomials

    A convergent expansion of the Airy's integral with incomplete Gamma functions

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    There are two main power series for the Airy functions, namely the Maclaurin and the asymptotic expansions. The former converges for all finite values of the complex variable, zz, but it requires a large number of terms for large values of z|z|, and the latter is a Poincar\'{e}-type expansion which is well-suited for such large values and where optimal truncation is possible. The asymptotic series of the Airy function shows a classical example of the Stokes phenomenon where a type of discontinuity occurs for the homonymous multipliers. A new series expansion is presented here that stems from the method of steepest descents, as can the asymptotic series, but which is convergent for all values of the complex variable. It originates in the integration of uniformly convergent power series representing the integrand of the Airy's integral in different sections of the integration path. The new series expansion is not a power series and instead relies on the calculation of complete and incomplete Gamma functions. In this sense, it is related to the Hadamard expansions. It is an alternative expansion to the two main aforementioned power series that also offers some insight into the transition zone for the Stokes' multipliers due to the splitting of the integration path. Unlike the Hadamard series, it relies on only two different expansions, separated by a branch point, one of which is centered at infinity. The interest of the new series expansion is mainly a theoretical one in a twofold way. First of all, it shows how to convert an asymptotic series into a convergent one, even if the rate of convergence may be slow for small values of z|z|. Secondly, it sheds some light on the Stokes phenomenon for the Airy function by showing the transition of the integration paths at argz=±2π/3\arg z = \pm 2 \pi/3.Comment: 21 pages, 23 figures. Changes in version 2: i) Footnote 10 has been added, ii) Figure 5 has been added for a deeper analysis of the results, iii) Reference 15 has been added, iv) Typo: A ±\pm was missing in argz=±2π/3\arg z = \pm 2 \pi/3 (abstract), v) Some font size changes and improved labelling in the figures Changes in version 3: minor edition change
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