arXiv subjects

Á. Baricz

Publications and source records attributed to Á. Baricz.

3 recordsLinked to original sources

arXiv2026

Riccati type recursions for some infinite series involving zeros of Bessel functions of the first kind

Some infinite series involving the positive zeros of Bessel functions of the first kind are investigated. The motivation behind these series lies in quantum mechanical perturbation problems, in which energies and matrix elements of unperturbed states are expressible in terms of zeros of Bessel functions of the first kind. The existing approach by Pedersen and Urbanowicz for calculating these series involves the Thomas-Reiche-Kuhn sum rule, differential recurrences for powers of Bessel function ratios or application of Lommel polynomials. In this paper an alternative approach is provided: a recursive algorithm is proposed that theoretically can produce the infinite series values in question. Our method is relatively simple and rely on three main ingredients: the Mittag-Leffler expansion and Riccati differential equation for the quotient of Bessel functions of the first kind, as well as the Taylor series coefficients of these ratios. The technique employed in the paper could be useful to treat similar problems where infinite series of zeros of special functions is involved.

math.CA↗

arXiv2014

The radius of univalence of the reciprocal of a product of two analytic functions

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the open unit disk $\ID$ with the normalization $f(0)=0= f'(0)-1$ and ${\mathcal S}$ be the class of univalent functions from ${\mathcal A}$. In this paper, we consider radius of univalence of $F$ defined by $F(z)=z^{3}/(f(z)g(z))$, where $f$ and $g$ belong to some subclasses of ${\mathcal A}$ (for which $f(z)/z$ and $g(z)/z$ are non-vanishing in $\ID$) and, in some cases in precise form, belonging to some subclasses of ${\mathcal S}$. All the results are proved to be sharp. Applications of our investigation through Bessel functions are also presented.

math.CV↗

arXiv2014

Modified Dini functions: monotonicity patterns and functional inequalities

We deduce some new functional inequalities, like Turán type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.

math.CA↗