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Inequalities for integrals of modified Bessel functions and expressions involving them
Simple inequalities are established for some integrals involving the modified
Bessel functions of the first and second kind. In most cases, we show that we
obtain the best possible constant or that our bounds are tight in certain
limits. We apply these inequalities to obtain uniform bounds for several
expressions involving integrals of modified Bessel functions. Such expressions
occur in Stein's method for variance-gamma approximation, and the results
obtained in this paper allow for technical advances in the method. We also
present some open problems that arise from this research.Comment: 20 pages. Final version. To appear in Journal of Mathematical
Analysis and Application
A generalization of determinant formulas for the solutions of Painlev\'e II and XXXIV equations
A generalization of determinant formulas for the classical solutions of
Painlev\'e XXXIV and Painlev\'e II equations are constructed using the
technique of Darboux transformation and Hirota's bilinear formalism. It is
shown that the solutions admit determinant formulas even for the transcendental
case.Comment: 20 pages, LaTeX 2.09(IOP style), submitted to J. Phys.
Multi-Dimensional Sigma-Functions
In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of -functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.Comment: 267 pages, 4 figure
Closed-form formulae for the derivatives of trigonometric functions at rational multiples of
In this sequel to our recent note it is shown, in a unified manner, by making
use of some basic properties of certain special functions, such as the Hurwitz
zeta function, Lerch zeta function and Legendre chi function, that the values
of all derivatives of four trigonometric functions at rational multiples of
can be expressed in closed form as simple finite sums involving the
Bernoulli and Euler polynomials. In addition, some particular cases are
considered.Comment: 5 page
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