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Rahul Kumar  One of the best experts on this subject based on the ideXlab platform.

a generalized modified Bessel Function and explicit transformations of certain lambert series
arXiv: Number Theory, 2020CoAuthors: Atul Dixit, Aashita Kesarwani, Rahul KumarAbstract:An exact transformation, which we call a \emph{master identity}, is obtained for the series $\sum_{n=1}^{\infty}\sigma_{a}(n)e^{ny}$ for $a\in\mathbb{C}$ and Re$(y)>0$. As corollaries when $a$ is an odd integer, we derive the wellknown transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta Function as well as Ramanujan's famous formula for $\zeta(2m+1)$. Corresponding new transformations when $a$ is a nonzero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan's formula for $\zeta(2m+1)$. Although not modular, it is surprising that such explicit transformations exist. The WigertBellman identity arising from the $a=0$ case of the master identity is derived too. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel Function $K_{\nu}(z)$. Koshliakov's integral evaluation is proved for the first time. It is then generalized using a wellknown kernel of Watson to obtain an interesting twovariable generalization of the modified Bessel Function. This generalization allows us to obtain a new transformation involving the sumsofsquares Function $r_k(n)$. Some results on Functions selfreciprocal in the Watson kernel are also obtained.

the generalized modified Bessel Function and its connection with voigt line profile and humbert Functions
Advances in Applied Mathematics, 2020CoAuthors: Rahul KumarAbstract:Abstract Recently Dixit, Kesarwani, and Moll introduced a generalization K z , w ( x ) of the modified Bessel Function K z ( x ) and showed that it satisfies an elegant theory similar to that of K z ( x ) . In this paper, we show that while K 1 2 ( x ) is an elementary Function, K 1 2 , w ( x ) can be written in the form of an infinite series of Humbert Functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta Function η ( z ) . We also establish a connection between K 1 2 , w ( x ) and the cumulative distribution Function corresponding to the Voigt line profile.

the generalized modified Bessel Function k_ z w x at z 1 2 and humbert Functions
arXiv: Classical Analysis and ODEs, 2018CoAuthors: Rahul KumarAbstract:Recently Dixit, Kesarwani, and Moll introduced a generalization $K_{z,w}(x)$ of the modified Bessel Function $K_{z}(x)$ and showed that it satisfies an elegant theory similar to $K_{z}(x)$. In this paper, we show that while $K_{\frac{1}{2}}(x)$ is an elementary Function, $K_{\frac{1}{2},w}(x)$ can be written in the form of an infinite series of Humbert Functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta Function $\eta(z)$.
Nico M. Temme  One of the best experts on this subject based on the ideXlab platform.

a generalized modified Bessel Function and a higher level analogue of the theta transformation formula
arXiv: Number Theory, 2017CoAuthors: Atul Dixit, Victor H Moll, Aashita Kesarwani, Nico M. TemmeAbstract:A new generalization of the modified Bessel Function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differentialdifference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a Function which gives a new pair of Functions reciprocal in the Koshliakov kernel $\cos \left( {{\pi z}} \right){M_{2z}}(4\sqrt {x} )  \sin \left( {{\pi z}} \right){J_{2z}}(4\sqrt {x} )$ and which subsumes the selfreciprocal pair involving $K_{z}(x)$. Its application towards finding modulartype transformations of the form $F(z, w, \alpha)=F(z,iw,\beta)$, where $\alpha\beta=1$, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the Functional equation of a nonholomorphic Eisenstein series on $SL_{2}(\mathbb{Z})$. This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann $\Xi$Function and consisting of a sum of products of two confluent hypergeometric Functions.

evaluation of the modified Bessel Function of the third kind of imaginary orders
Journal of Computational Physics, 2002CoAuthors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The evaluation of the modified Bessel Function of the third kind of purely imaginary order Kia(x) is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions, and a continued fraction method, depending on the ranges of x and a. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method, and the nonoscillatory integral representation can be used to accurately compute the Function Kia(x) in the range 0 ≤ a ≤ 200, 0 ≤ x ≤ 100; using a similar scheme the derivative K'ia(x) can also be computed within these ranges.

evaluation of the modified Bessel Function of the third kind of imaginary orders
Report Modelling Analysis and Simulation, 2001CoAuthors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The evaluation of the modified Bessel Function of the third kind of purely imaginary order $K_{ia}(x)$ is discussed; we also present analogous results for the derivative. The methods are based on the use of Maclaurin series, nonoscillatory integral representations, asymptotic expansions and a continued fraction method, depending on the ranges of $x$ and $a$. We discuss the range of applicability of the different approaches considered and conclude that power series, the continued fraction method and the nonoscillatory integral representation can be used to accurately compute the Function $K_{ia}(x)$ in the range $0le ale 200$, $0le xle 100$; using a similar scheme the derivative $K_{ia}^{prime}(x)$ can also be computed within these ranges.
Roger Grinter  One of the best experts on this subject based on the ideXlab platform.

characterizing emitted electromagnetic radiation the vector spherical harmonic Bessel Function description
Journal of Physics B, 2014CoAuthors: Roger GrinterAbstract:General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vectorsphericalharmonic/BesselFunction description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated.
A Sotomayor  One of the best experts on this subject based on the ideXlab platform.

quasi rational analytic approximation for the modified Bessel Function i1 x with high accuracy
Symmetry, 2021CoAuthors: Pablo Martin, Jorge Olivares, Eduardo Rojas, A SotomayorAbstract:A new simple and accurate expression to approximate the modified Bessel Function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multipoint quasirational approximation technique, MPQA. This method uses the power series of the Bessel Function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting Function. The fitting expression is formed by elementary Functions combined with rational ones. In the present work, a sum of hyperbolic Functions was selected as elementary Functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation Function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 10−2 to 4×10−4, opening the possibility of applying the new improved method to other Bessel Functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument.

precise analytic approximation for the modified Bessel Function i_1 x
Revista Mexicana De Fisica, 2017CoAuthors: Pablo Martin, Jorge Olivares, A SotomayorAbstract:Precise analytic approximations have been found for the modified Bessel Function $I_1(x)$ of order one. The approximant is simple and with good accuracy for every positive value of the variable $x$. The method to obtain the approximant is an improvement of the multipoint quasirational approximation technique, MPQA. This is obtained combining rational Functions with elementary auxiliary Functions as a bridge Function between the power series and asymptotic expansion.

multi point quasi rational approximants for the modified Bessel Function i1 x
Journal of Physics: Conference Series, 2016CoAuthors: P Martin, J Olivares, L Cortesvega, A SotomayorAbstract:Approximants for the modified Bessel Function I1(x) has been found using the multipoint quasirational technique. The approximations here determined has good accuracy for any positive value of the variable, and it seems to be adequate for most of the works where this Function are used. Furthermore, the approximants are simple to calculate numerically in a direct way or using any usual MAPLE or MATLAB software.
R B Paris  One of the best experts on this subject based on the ideXlab platform.

on the nu zeros of the modified Bessel Function k_ i nu x of positive argument
arXiv: Classical Analysis and ODEs, 2021CoAuthors: R B ParisAbstract:The modified Bessel Function of the second kind $K_{i\nu}(x)$ of imaginary order for fixed $x>0$ possesses a countably infinite sequence of real zeros. Recently it has been shown that the $n$th zero behaves like $\nu_n\sim \pi n/\log\,n$ as $n\to\infty$. In this note we determine a more precise estimate for the bahaviour of these zeros for large $n$ by making use of the known asymptotic expansion of $K_{i\nu}(x)$ for large $\nu$. Numerical results are presented to illustrate the accuracy of the expansion obtained.

the asymptotics of the generalised Bessel Function
Mathematica Æterna, 2017CoAuthors: R B ParisAbstract:We demonstrate how the asymptotics for large z of the generalised Bessel Function 0Ψ1(z) = X∞ n=0 z n Γ(an + b)n! , where a > −1 and b is any number (real or complex), may be obtained by exploiting the wellestablished asymptotic theory of the generalised Wright Function pΨq(z). A summary of this theory is given and an algorithm for determining the coefficients in the associated exponential expansions is discussed in an appendix. We pay particular attention to the case a = − 1 2 , where the expansion for z → ±∞ consists of an exponentially small contribution that undergoes a Stokes phenomenon. We also examine the different nature of the asymptotic expansions as a Function of arg z when −1 < a < 0, taking into account the Stokes phenomenon that occurs on the rays arg z = 0 and arg z = ±π(1 + a) for the associated Function 1Ψ0(z). These regions are more precise than those given by Wright in his 1940 paper. Numerical computations are carried out to verify several of the expansions developed in the paper.

a note on a modified Bessel Function integral
arXiv: Classical Analysis and ODEs, 2015CoAuthors: R B ParisAbstract:We investigate the integral \[\int_0^\infty \cosh^\mu\!t\,K_\nu(z\cosh t)\,dt \qquad \Re(z)>0,\] where $K$ denotes the modified Bessel Function, for nonnegative integer values of the parameters $\mu$ and $\nu$. When the integers are of different parity, closedform expressions are obtained in terms of $z^{1}e^{z}$ multiplied by a polynomial in $z^{1}$ of degree dependent on the sign of $\mu\nu$.