arXiv subjects

Sanjeev Singh

Publications and source records attributed to Sanjeev Singh.

14 recordsLinked to original sources

arXiv2026

Modified Halley's method for computation of zeros of solution of second order ODEs

This paper develops an efficient iterative method for computing all zeros of solutions of second order ordinary differential equations. A third order Halleys method is first derived by approximating the solution of an associated Riccati differential equation. To improve computational efficiency, a modified Halleys method is proposed by fixing one of the functions in Halleys scheme as a constant. The modified Halleys method also retains third order convergence. Based on the behavior of the coefficients of the second order ODE, nonlocal convergence results are established for both Halleys and modified Halleys methods. Suitable initial guesses for computing all zeros of solutions of second order ODEs in a given interval are also presented for both methods. Furthermore, algorithms based on the modified Halleys method are developed for to compute all nodes and weights for Gauss Legendre and Gauss Hermite quadratures. A comparative numerical study with recent methods demonstrates the efficiency of the proposed algorithms.

math.NA↗

arXiv2026

Efficient third-order iterative algorithms for computing zeros of special functions

This manuscript presents a novel and reliable third-order iterative procedure for computing the zeros of solutions to second-order ordinary differential equations. By approximating the solution of the related Riccati differential equation using the trapezoidal rule, this study has derived the proposed third-order method. This work establishes sufficient conditions to ensure the theoretical non-local convergence of the proposed method. This study provides suitable initial guesses for the proposed third-order iterative procedure to compute all zeros in a given interval of the solutions to second-order ordinary differential equations. The orthogonal polynomials like Legendre and Hermite, as well as the special functions like Bessel, Coulomb wave, confluent hypergeometric, and cylinder functions, satisfy the proposed conditions for convergence. Numerical simulations demonstrate the effectiveness of the proposed theory. This work also presents a comparative analysis with recent studies.

math.NA↗

arXiv2025

Bounds and asymptotic expansions for the radii of convexity and uniform convexity of normalized Bessel functions

This paper explores the asymptotic behaviour of the radii of convexity and uniform convexity for normalized Bessel functions with respect to large order. We provide detailed asymptotic expansions for these radii and establish recurrence relations for the associated coefficients. Additionally, we derive generalized bounds for the radii of convexity and uniform convexity by applying the Euler-Rayleigh inequality and potential polynomials. The asymptotic inversion method and Rayleigh sums are the main tools used in the proofs.

math.CV↗

arXiv2025

The radius of starlikeness of regular Coulomb wave functions

Motivated by the pioneering work of M.S. Robertson [Ro54] and R.K. Brown [Br60], [Br62], who examined the geometric properties of some normalised solutions of second-order homogeneous differential equations, in this paper we investigate the radii of univalence and starlikeness for two kind of normalised regular Coulomb wave functions. Moreover, a generalized normalised Bessel function is introduced, and its radius of starlikeness is studied by using two different approaches. In addition, the asymptotic behaviour, with respect to the large order, of the radius of starlikeness of one type of normalised Coulomb wave functions is considered, which is in fact the first zero of the derivative of the regular Coulomb wave function. We derive a complete asymptotic expansion for this radius of starlikeness and provide a recurrence relation for the coefficients of this expansion. The proof is based on Rayleigh sums of the zeros of Coulomb wave functions, asymptotic inversion and some basic results on regular Coulomb wave functions developed by \v{S}tampach and \v{S}\'{t}ov\'{\i}\v{c}ek [SS14].

math.CA↗

arXiv2024

Infinitely divisible modified Bessel distributions

In this paper we focus on continuous univariate probability distributions, like McKay distributions, $K$-distribution, generalized inverse Gaussian distribution and generalised McKay distributions, with support $[0,\infty),$ which are related to modified Bessel functions of the first and second kinds and in most cases we show that they belong to the class infinitely divisible distributions, self-decomposable distributions, generalized gamma convolutions and hyperbolically completely monotone densities. Some of the results are known, however the proofs are new and we use special functions technique. Integral representations of quotients of Tricomi hypergeometric functions as well as of quotients of Gaussian hypergeometric functions, or modified Bessel functions of the second kind play an important role in our study. In addition, by using a different approach we rediscover a Stieltjes transform representation due to Hermann Hankel for the product of modified Bessel functions of the first and second kinds and we also deduce a series of new Stieltjes transform representations for products, quotients and their reciprocals concerning modified Bessel functions of the first and second kinds. By using these results we obtain new infinitely divisible modified Bessel distributions with Laplace transforms related to modified Bessel functions of the first and second kind. Moreover, we show that the new Stieltjes transform representations have some interesting applications and we list some open problems which may be of interest for further research. In addition, we present a new proof via the Pick function characterization theorem for the infinite divisibility of the ratio of two gamma random variables and we present some new Stieltjes transform representations of quotients of Tricomi hypergeometric functions.

math.PR↗

arXiv2023

Can I say, now machines can think?

Generative AI techniques have opened the path for new generations of machines in diverse domains. These machines have various capabilities for example, they can produce images, generate answers or stories, and write codes based on the "prompts" only provided by users. These machines are considered 'thinking minds' because they have the ability to generate human-like responses. In this study, we have analyzed and explored the capabilities of artificial intelligence-enabled machines. We have revisited on Turing's concept of thinking machines and compared it with recent technological advancements. The objections and consequences of the thinking machines are also discussed in this study, along with available techniques to evaluate machines' cognitive capabilities. We have concluded that Turing Test is a critical aspect of evaluating machines' ability. However, there are other aspects of intelligence too, and AI machines exhibit most of these aspects.

cs.AI↗

arXiv2020

A Survey on Software Defined Networking: Architecture for Next Generation Network

The evolution of software defined networking (SDN) has played a significant role in the development of next-generation networks (NGN). SDN as a programmable network having service provisioning on the fly has induced a keen interest both in academic world and industry. In this article, a comprehensive survey is presented on SDN advancement over conventional network. The paper covers historical evolution in relation to SDN, functional architecture of the SDN and its related technologies, and OpenFlow standards/protocols, including the basic concept of interfacing of OpenFlow with network elements (NEs) such as optical switches. In addition a selective architecture survey has been conducted. Our proposed architecture on software defined heterogeneous network, points towards new technology enabling the opening of new vistas in the domain of network technology, which will facilitate in handling of huge internet traffic and helps infrastructure and service providers to customize their resources dynamically. Besides, current research projects and various activities as being carried out to standardize SDN as NGN by different standard development organizations (SODs) have been duly elaborated to judge how this technology moves towards standardization.

cs.NI↗

arXiv2018

Geometric and monotonic properties of hyper-Bessel functions

Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-Pólya class of real entire functions, intensively.

math.CV↗

arXiv2017

Zeros of some special entire functions

The real and complex zeros of some special entire functions such as Wright, hyper-Bessel, and a special case of generalized hypergeometric functions are studied by using some classical results of Laguerre, Obreschkhoff, Pólya and Runckel. The obtained results extend the known theorem of Hurwitz on exact number of nonreal zeros of Bessel functions of the first kind. Moreover, results on zeros of derivatives of Bessel functions and cross-product of Bessel functions are also given, which are related to some recent open problems.

math.CA↗

arXiv2015

Bounds for the product of modified Bessel functions

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of first and second kind. Certain bounds for the product of modified Bessel functions of first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of first and second kind of order zero. A new Turán type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research.

math.CA↗

arXiv2015

Cross-product of Bessel functions: monotonicity patterns and functional inequalities

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently developed infinite product representations for Dini functions and cross-product of Bessel functions.

math.CA↗

arXiv2015

Turán type inequalities for general Bessel functions

In this paper some Turán type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Turánian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.

math.CA↗

arXiv2014

Turán type inequalities for confluent hypergeometric functions of the second kind

In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor $F$ distribution.

math.CA↗

arXiv2014

Turán type inequalities for Struve functions

Some Turán type inequalities for Struve functions of the first kind are deduced by using various methods developed in the case of Bessel functions of the first and second kind. New formulas, like Mittag-Leffler expansion, infinite product representation for Struve functions of the first kind, are obtained, which may be of independent interest. Moreover, some complete monotonicity results and functional inequalities are deduced for Struve functions of the second kind. These results complement naturally the known results for a particular case of Lommel functions of the first kind, and for modified Struve functions of the first and second kind.

math.CA↗