Computing the Kummer function U(a,b,z)U(a,b,z) for small values of the arguments

We describe methods for computing the Kummer function $U(a,b,z)$ for small values of $z$, with special attention to small values of $b$. For these values of $b$ the connection formula that represents $U(a,b,z)$ as a linear combination of two ${}_1F_1$-functions needs a limiting procedure. We use the power series of the ${}_1F_1$-functions and consider the terms for which this limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we consider the derivative $U^\prime(a,b,z)$ as well. We also discuss the performance for small $\vert z\vert$ of an asymptotic approximation of the Kummer function in terms of modified Bessel functions

Conical: an extended module for computing a numerically satisfactory pair of solutions of the differential equation for conical functions

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine for computing the function ${{\rm R}}^{m}_{-\frac{1}{2}+i\tau}(x)$, a real-valued numerically satisfactory companion of the function ${\rm P}^m_{-\tfrac12+i\tau}(x)$ for $x>1$. In this way, a natural basis for solving Dirichlet problems bounded by conical domains is provided.Comment: To appear in Computer Physics Communication

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

Computing hypergeometric functions rigorously

We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions ${}_0F_1$, ${}_1F_1$, ${}_2F_1$ and ${}_2F_0$ (or the Kummer $U$-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function ${}_pF_q$ and computation of high-order parameter derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case E-G in section 8.5 (table 6, figure 2); adjusted paper siz

The unpolarized two-loop massive pure singlet Wilson coefficients for deep-inelastic scattering

We calculate the massive two--loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless two--loop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincar\'e letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions $F_2$ and $F_L$ we also derive improved asymptotic representations adding power corrections. Numerical results are presented.Comment: 42, pages Latex, 8 Figure

On the dust abundance gradients in late-type galaxies: I. Effects of destruction and growth of dust in the interstellar medium

We present basic theoretical constraints on the effects of destruction by supernovae (SNe) and growth of dust grains in the interstellar medium (ISM) on the radial distribution of dust in late-type galaxies. The radial gradient of the dust-to-metals ratio is shown to be essentially flat (zero) if interstellar dust is not destroyed by SN shock waves and all dust is produced in stars. If there is net dust destruction by SN shock waves, the dust-to-metals gradient is flatter than or equal to the metallicity gradient (assuming the gradients have the same sign). Similarly, if there is net dust growth in the ISM, then the dust-to-metals gradient is steeper than or equal to the metallicity gradient. The latter result implies that if dust gradients are steeper than metallicity gradients, i.e., the dust-to-metals gradients are not flat, then it is unlikely dust destruction by SN shock waves is an efficient process, while dust growth must be a significant mechanism for dust production. Moreover, we conclude that dust-to-metals gradients can be used as a diagnostic for interstellar dust growth in galaxy discs, where a negative slope indicates dust growth.Comment: 12 pages, 4 figures. Accepted for publication in MNRA

On computing some special values of hypergeometric functions

The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in this paper we continue the path of research started in two our previous papers appeared on [30] and [31] providing some contribution to such functions computability inside and outside their disk of convergence. This is accomplished through two different approaches. The first is to provide some new results in the spirit of theorem 3.1 of 31] obtaining formulae for multivariable hypergeometric functions by generalizing a well known Kummer's identity to the hypergeometric functions of two or more variable like those of Appell and Lauricella.Comment: 21 pages. Sixth version. To appear in Journal of Mathematical Analysis and Application