The Experts below are selected from a list of 2304 Experts worldwide ranked by ideXlab platform
Nico M. Temme - One of the best experts on this subject based on the ideXlab platform.
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a generalized Modified Bessel Function and a higher level analogue of the theta transformation formula
arXiv: Number Theory, 2017Co-Authors: 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 differential-difference 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 self-reciprocal pair involving $K_{z}(x)$. Its application towards finding modular-type 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 non-holomorphic 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.
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on Modified asymptotic series involving confluent hypergeometric Functions
Electronic Transactions on Numerical Analysis, 2009Co-Authors: Alfredo Deano, Nico M. TemmeAbstract:A modification of the Poincare-type asymptotic expansion f or Functions defined by Laplace trans- forms is analyzed. This modification is based on an alternati ve power series expansion of the integrand, and the convergence properties are seen to be superior to those of th e original asymptotic series. The resulting Modified asymptotic expansion involves a series of confluent hyperge ometric Functions U(a,c,z), which can be computed by means of continued fractions in a backward recursion scheme. Numerical examples are included, such as the incomplete gamma Function ( a, z) and the Modified Bessel Function K�(z) for large values of z. It is observed that the same procedure can be applied to uniform asymptotic expansions when extra parameters become large as well.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Journal of Computational and Applied Mathematics, 2003Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders (Kia(x)) near the transition point x = a, is discussed. In A. Gil et al., Evaluation of the Modified Bessel Functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398-411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
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evaluation of the Modified Bessel Function of the third kind of imaginary orders
Journal of Computational Physics, 2002Co-Authors: 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.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Report Modelling Analysis and Simulation, 2002Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders ($K_{ia}(x)$) near the transition point $x=a$, is discussed. In [2], an algorithm for the evaluation of $K_{ia}(x)$ was presented, which made use of series, a continued fraction method and non-oscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
Amparo Gil - One of the best experts on this subject based on the ideXlab platform.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Journal of Computational and Applied Mathematics, 2003Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders (Kia(x)) near the transition point x = a, is discussed. In A. Gil et al., Evaluation of the Modified Bessel Functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398-411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
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evaluation of the Modified Bessel Function of the third kind of imaginary orders
Journal of Computational Physics, 2002Co-Authors: 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.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Report Modelling Analysis and Simulation, 2002Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders ($K_{ia}(x)$) near the transition point $x=a$, is discussed. In [2], an algorithm for the evaluation of $K_{ia}(x)$ was presented, which made use of series, a continued fraction method and non-oscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
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evaluation of the Modified Bessel Function of the third kind of imaginary orders
Report Modelling Analysis and Simulation, 2001Co-Authors: 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, non-oscillatory 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 non-oscillatory 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.
Victor H Moll - One of the best experts on this subject based on the ideXlab platform.
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a generalized Modified Bessel Function and a higher level analogue of the theta transformation formula
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Atul Dixit, Aashita Kesarwani, Victor H MollAbstract:Abstract 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 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 ( π z ) M 2 z ( 4 x ) − sin ( π z ) J 2 z ( 4 x ) and which subsumes the self-reciprocal pair involving K z ( x ) . Its application towards finding modular-type transformations of the form F ( z , w , α ) = F ( z , i w , β ) , where α β = 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 non-holomorphic Eisenstein series on S L 2 ( 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 Ξ-Function and consisting of a sum of products of two confluent hypergeometric Functions.
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a generalized Modified Bessel Function and a higher level analogue of the theta transformation formula
arXiv: Number Theory, 2017Co-Authors: 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 differential-difference 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 self-reciprocal pair involving $K_{z}(x)$. Its application towards finding modular-type 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 non-holomorphic 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.
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self reciprocal Functions powers of the riemann zeta Function and modular type transformations
Journal of Number Theory, 2015Co-Authors: Atul Dixit, Victor H MollAbstract:Abstract Integrals containing the first power of the Riemann Ξ-Function as part of the integrand that lead to modular-type transformations have been previously studied by Ramanujan, Hardy, Koshlyakov, Ferrar and others. An integral containing the square of the Riemann Ξ-Function and involving an extra parameter z, whose type naturally extends that of the afore-mentioned integrals, was studied by Ramanujan. This integral implicitly involves squaring of the Functional equation of ζ ( s ) . A unifying procedure to analyze general integrals of this type is studied here along with the interesting modular transformations that they generate. This also includes generalization of some transformations of Koshlyakov involving a series containing the Modified Bessel Function K 0 ( x ) .
Javier Segura - One of the best experts on this subject based on the ideXlab platform.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Journal of Computational and Applied Mathematics, 2003Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders (Kia(x)) near the transition point x = a, is discussed. In A. Gil et al., Evaluation of the Modified Bessel Functions of the third kind of imaginary orders, J. Comput. Phys. 17 (2002) 398-411, an algorithm for the evaluation of Kia(x) was presented, which made use of series, a continued fraction method and nonoscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
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evaluation of the Modified Bessel Function of the third kind of imaginary orders
Journal of Computational Physics, 2002Co-Authors: 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.
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computation of the Modified Bessel Function of the third kind of imaginary orders uniform airy type asymptotic expansion
Report Modelling Analysis and Simulation, 2002Co-Authors: Amparo Gil, Javier Segura, Nico M. TemmeAbstract:The use of a uniform Airy-type asymptotic expansion for the computation of the Modified Bessel Functions of the third kind of imaginary orders ($K_{ia}(x)$) near the transition point $x=a$, is discussed. In [2], an algorithm for the evaluation of $K_{ia}(x)$ was presented, which made use of series, a continued fraction method and non-oscillating integral representations. The range of validity of the algorithm was limited by the singularity of the steepest descent paths near the transition point. We show how uniform Airy-type asymptotic expansions fill the gap left by the steepest descent method.
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evaluation of the Modified Bessel Function of the third kind of imaginary orders
Report Modelling Analysis and Simulation, 2001Co-Authors: 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, non-oscillatory 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 non-oscillatory 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.
Atul Dixit - One of the best experts on this subject based on the ideXlab platform.
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a generalized Modified Bessel Function and explicit transformations of certain lambert series
arXiv: Number Theory, 2020Co-Authors: 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 well-known 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 non-zero 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 Wigert-Bellman 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 well-known kernel of Watson to obtain an interesting two-variable generalization of the Modified Bessel Function. This generalization allows us to obtain a new transformation involving the sums-of-squares Function $r_k(n)$. Some results on Functions self-reciprocal in the Watson kernel are also obtained.
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a generalized Modified Bessel Function and a higher level analogue of the theta transformation formula
Journal of Mathematical Analysis and Applications, 2018Co-Authors: Atul Dixit, Aashita Kesarwani, Victor H MollAbstract:Abstract 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 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 ( π z ) M 2 z ( 4 x ) − sin ( π z ) J 2 z ( 4 x ) and which subsumes the self-reciprocal pair involving K z ( x ) . Its application towards finding modular-type transformations of the form F ( z , w , α ) = F ( z , i w , β ) , where α β = 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 non-holomorphic Eisenstein series on S L 2 ( 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 Ξ-Function and consisting of a sum of products of two confluent hypergeometric Functions.
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a generalized Modified Bessel Function and a higher level analogue of the theta transformation formula
arXiv: Number Theory, 2017Co-Authors: 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 differential-difference 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 self-reciprocal pair involving $K_{z}(x)$. Its application towards finding modular-type 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 non-holomorphic 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.
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self reciprocal Functions powers of the riemann zeta Function and modular type transformations
Journal of Number Theory, 2015Co-Authors: Atul Dixit, Victor H MollAbstract:Abstract Integrals containing the first power of the Riemann Ξ-Function as part of the integrand that lead to modular-type transformations have been previously studied by Ramanujan, Hardy, Koshlyakov, Ferrar and others. An integral containing the square of the Riemann Ξ-Function and involving an extra parameter z, whose type naturally extends that of the afore-mentioned integrals, was studied by Ramanujan. This integral implicitly involves squaring of the Functional equation of ζ ( s ) . A unifying procedure to analyze general integrals of this type is studied here along with the interesting modular transformations that they generate. This also includes generalization of some transformations of Koshlyakov involving a series containing the Modified Bessel Function K 0 ( x ) .