Related papers: On determinants of modified Bessel functions and e…
We consider a 3-parameter family of linear special double confluent Heun equations introduced and studied by V.M.Buchstaber and S.I.Tertychnyi, which is an equivalent presentation of a model of Josephson junction in superconductivity.…
We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{\lambda,\mu,n}$ is a family of differential operators of order two. They depend on complex parameters $\lambda$, $\mu$, $n$. Its…
The explicit formulas for the maps interconnecting the sets of solutions of the special double confluent Heun equation and the equation of the RSJ model of overdamped Josephson junction in case of shifted sinusoidal bias are given. The…
A family $\mathcal{T}^{(\nu)}$, $\nu\in\mathbb{R}$, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton $q$-difference equation. The corresponding matrix operators defined on the linear hull…
The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V.M.Buchstaber and S.I.Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in…
We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent…
It is known that Struve function $\mathbf H_\nu$ and modified Struve function $\mathbf L_\nu$ are closely connected to Bessel function of the first kind $J_\nu$ and to modified Bessel function of the first kind $I_\nu$ and possess…
A rectangular matrix is called totally positive, if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called strictly totally positive, if one can normalize its…
Mathieu ordinary differential equation is of Fuchsian types with the two regular and one irregular singularities. In contrast, Heun equation of Fuchsian types has the four regular singularities. Heun equation has the four kind of confluent…
We present an exact mathematical description of Josephson vortices and of the Meissner effect in periodic thin-layer superconductor/insulator structures with an arbitrary number of identical junctions N-1 (N is the number of superconducting…
We examine convergent representations for the sum of Bessel functions \[\sum_{n=1}^\infty \frac{J_\mu(na) J_\nu(nb)}{n^{\alpha}}\] for $\mu$, $\nu\geq0$ and positive values of $a$ and $b$. Such representations enable easy computation of the…
We derive two distinct asymptotic expansions for the zeros $j_{\nu,k}^{(n)}$ of the $n$-th derivative of Bessel function $J_\nu^{(n)}(x)$. The first is a McMahon-type expansion for the case when $k \to \infty$ with fixed $\nu$, for which we…
The $\nu$-zeros of the Bessel functions of purely imaginary order are examined for fixed argument $x>0$. In the case of the modified Bessel function of the second kind $K_{i\nu}(x)$, it is known that it possesses a countably infinite…
We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for…
In earlier work, we introduced three families of polynomials where the generating function of each set includes one of the three Jackson $q$-analogs of the Bessel function. This paper gives determinant representation for each family, their…
This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to…
A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby…
In this paper we study the following Bessel series $\sum _{l=1}^{\infty } {J_{l+m'}(r)J_{l+m}(r)}{(l+\beta)^\alpha}$ for any $m,m'\in\mathbb{Z}$, $\alpha\in\mathbb{R}$ and $\beta>-1$. They are a particular case of the second type Neumann…