CP methods and the evaluation of negative energy Coulomb Whittaker functions

The CP (constant reference potential perturbation) methods are specialized methods for the solution of Schrödinger equations. They allow big step sizes and are well suited for oscillatory problems. As an illustration of the power of these methods, oscillating Whittaker functions of the second kind are obtained with high accuracy, using a high-order CP-method.     

Zeros of the Macdonald function of complex order

The z  -zeros of the modified Bessel function of the third kind _Kν(z)$K_{\nu }(z)$, also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν$\nu$. Approximate expressions for the zeros, applicable in the cases of very small or very large |ν|$\left|\nu \right|$, are given. The behaviour of the zeros for varying |ν|$\left|\nu \right|$ or argν$\arg \nu$, obtained numerically, is illustrated by means of some graphics.     

Zeros of a function given by the series of functions

We present the numerical analysis on computing zeros of a function given by the series of functions. In fact, these zeros are Dirichlet eigenvalues of a geodesic ball in the n-sphere. Furthermore, some numerical results from this analysis are included.     

Whittaker方程对非完整力学系统的推广

1904年Whittaker利用能量积分将一个完整保守力学系统问题降阶为一个带有较少自由度系统问题.并得到了Whittaker方程[1].本文推导对于非完整力学系统的这类方程.并称之为广义Whittaker方程;然后把这些方程变换为Nielsen形式;最后举例说明新方程的应用.     

Orthogonal polynomials associated with Coulomb wave functions

A new class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials have in the theory of Bessel functions. The orthogonality measure for this new class is described in detail. In addition, the orthogonality measure problem is discussed on a more general level. Apart from this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of certain formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix _JL$J_L$ whose eigenvalues coincide with the reciprocal values of the zeros of the regular Coulomb wave function _FL(η,ρ)$F_L(\eta ,\rho )$. The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.     

Zeros on the critical line of Dirichlet series associated with Hilbert modular forms

The relation is studied between the behavior of Hilbert modular forms in a neighborhood of some manifold and the zeros on the critical line of the associated Dirichlet series. The results are applied to Eisenstein series associated with quadratic forms over real quadratic fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 89–123, 1978.     

Zeros of series with respect to the Rademacher system

Mathematical Notes -     

Whittaker Modules for the Insertion-Elimination Lie Algebra

This paper addresses the representation theory of the insertion-elimination Lie algebra, a Lie algebra that can be naturally realized in terms of tree-inserting and tree-eliminating operations on rooted trees. The insertion-elimination algebra admits a triangular decomposition in the sense of Moody and Pianzola, and thus it is natural to define Whittaker modules corresponding to a given algebra homomorphism. Among other results, we show that the standard Whittaker modules are simple under certain constraints on the corresponding algebra homomorphism.     

Zeros of the Davenport-Heilbronn counterexample

We compute zeros off the critical line of a Dirichlet series considered by H. Davenport and H. Heilbronn. This computation is accomplished by deforming a Dirichlet series with a set of known zeros into the Davenport-Heilbronn series.

     

Zeros of the Riemann zeta-function

A proof that the Riemann zeta-function (+ it) has no zeros in the region where R=9.65 and T=12.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 419–429, October, 1970.