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Arpad Baricz - One of the best experts on this subject based on the ideXlab platform.
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bounds for radii of starlikeness of some q Bessel Functions
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Ibrahim Aktas, Arpad BariczAbstract:In this paper the radii of starlikeness of the Jackson and Hahn-Exton $q$-Bessel Functions are considered and for each of them three different normalization are applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these Functions tight lower and upper bounds for the radii of starlikeness of these Functions are obtained. The Laguerre-P\'olya class of real entire Functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.
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radii of starlikeness and convexity of some q Bessel Functions
Journal of Mathematical Analysis and Applications, 2016Co-Authors: Arpad Baricz, Dimitar K Dimitrov, Istvan MezoAbstract:Abstract Geometric properties of the Jackson and Hahn–Exton q -Bessel Functions are studied. For each of them, three different normalizations are applied in such a way that the resulting Functions are analytic in the unit disk of the complex plane. For each of the six Functions we determine the radii of starlikeness and convexity precisely by using their Hadamard factorization. These are q -generalizations of some known results for Bessel Functions of the first kind. The characterization of entire Functions from the Laguerre–Polya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing property of the zeros of Jackson and Hahn–Exton q -Bessel Functions and their derivatives is also useful in the proof of the main results. We also deduce a necessary and sufficient condition for the close-to-convexity of a normalized Jackson q -Bessel function and its derivatives. Some open problems are proposed at the end of the paper.
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the radius of alpha convexity of normalized Bessel Functions of the first kind
Computational Methods and Function Theory, 2016Co-Authors: Arpad Baricz, Halit Orhan, Robert SzaszAbstract:The radii of $$\alpha $$ -convexity are deduced for three different kinds of normalized Bessel Functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when $$\alpha \in [0,1]$$ , and they are decreasing with respect to the parameter $$\alpha $$ . The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel Functions of the first kind. The key tools in the proofs are some interlacing properties of the zeros of some Dini Functions and the zeros of Bessel Functions of the first kind.
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the radius of convexity of normalized Bessel Functions
Analysis Mathematica, 2015Co-Authors: Arpad Baricz, Robert SzaszAbstract:The radius of convexity of two normalized Bessel Functions of the first kind are determined in the case when the order is between -2 and -1. Our methods include the minimum principle for harmonic Functions, the Hadamard factorization of some Dini Functions, properties of the zeros of Dini Functions via Lommel polynomials and some inequalities for complex and real numbers. The results on the zeros of the combination of Bessel Functions of the first kind may be of independent interest.
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on an identity for zeros of Bessel Functions
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Tibor K. Pogány, Arpad Baricz, Dragana Jankov Masirevic, Robert SzaszAbstract:In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel Functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel Functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special Functions, like Struve Functions of the first kind, and modified Bessel Functions of the second kind.
Robert Szasz - One of the best experts on this subject based on the ideXlab platform.
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the radius of uniform convexity of Bessel Functions
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Erhan Deniz, Robert SzaszAbstract:Abstract In this paper, we determine the radius of uniform convexity for three kinds of normalized Bessel Functions of the first kind. In the cases considered the normalized Bessel Functions are uniformly convex on the determined disks. Moreover, necessary and sufficient conditions are given for the parameters of the three normalized Functions such that they to be uniformly convex in the open unit disk. The basic tool of this study is Bessel Functions in series.
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the radius of alpha convexity of normalized Bessel Functions of the first kind
Computational Methods and Function Theory, 2016Co-Authors: Arpad Baricz, Halit Orhan, Robert SzaszAbstract:The radii of $$\alpha $$ -convexity are deduced for three different kinds of normalized Bessel Functions of the first kind and it is shown that these radii are between the radii of starlikeness and convexity, when $$\alpha \in [0,1]$$ , and they are decreasing with respect to the parameter $$\alpha $$ . The results presented in this paper unify some recent results on the radii of starlikeness and convexity for normalized Bessel Functions of the first kind. The key tools in the proofs are some interlacing properties of the zeros of some Dini Functions and the zeros of Bessel Functions of the first kind.
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the radius of convexity of normalized Bessel Functions
Analysis Mathematica, 2015Co-Authors: Arpad Baricz, Robert SzaszAbstract:The radius of convexity of two normalized Bessel Functions of the first kind are determined in the case when the order is between -2 and -1. Our methods include the minimum principle for harmonic Functions, the Hadamard factorization of some Dini Functions, properties of the zeros of Dini Functions via Lommel polynomials and some inequalities for complex and real numbers. The results on the zeros of the combination of Bessel Functions of the first kind may be of independent interest.
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on an identity for zeros of Bessel Functions
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Tibor K. Pogány, Arpad Baricz, Dragana Jankov Masirevic, Robert SzaszAbstract:In this paper our aim is to present an elementary proof of an identity of Calogero concerning the zeros of Bessel Functions of the first kind. Moreover, by using our elementary approach we present a new identity for the zeros of Bessel Functions of the first kind, which in particular reduces to some other new identities. We also show that our method can be applied for the zeros of other special Functions, like Struve Functions of the first kind, and modified Bessel Functions of the second kind.
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the radius of convexity of normalized Bessel Functions
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Arpad Baricz, Robert SzaszAbstract:The radius of convexity of two normalized Bessel Functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic Functions, the Hadamard factorization of some Dini Functions, properties of the zeros of Dini Functions via Lommel polynomials and some inequalities for complex and real numbers.
Alexandru Zaharescu - One of the best experts on this subject based on the ideXlab platform.
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sums of squares and products of Bessel Functions
Advances in Mathematics, 2018Co-Authors: Bruce C Berndt, Sun Kim, Alexandru Zaharescu, Atul DixitAbstract:Abstract Let r k ( n ) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoi summation formula for r k ( n ) , k ≥ 2 , proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the Functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of r k ( n ) and a product of two Bessel Functions, and a series involving r k ( n ) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.
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sums of squares and products of Bessel Functions
arXiv: Number Theory, 2017Co-Authors: Bruce C Berndt, Sun Kim, Alexandru Zaharescu, Atul DixitAbstract:Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We rigorously prove for the first time a Voronoi summation formula for $r_k(n), k\geq2,$ proved incorrectly by A. I. Popov and later rediscovered by A. P. Guinand, but without proof and without conditions on the Functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of $r_k(n)$ and a product of two Bessel Functions, and a series involving $r_k(n)$ and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G. H. Hardy, and of A. L. Dixon and W. L. Ferrar, as well as of a classical result of A. I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.
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circle and divisor problems and double series of Bessel Functions
Advances in Mathematics, 2013Co-Authors: Bruce C Berndt, Sun Kim, Alexandru ZaharescuAbstract:Abstract In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel Functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.
Zachary Slepian - One of the best experts on this subject based on the ideXlab platform.
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an exact integral to sum relation for products of Bessel Functions
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2021Co-Authors: Oliver H E Philcox, Zachary SlepianAbstract:A useful identity relating the infinite sum of two Bessel Functions to their infinite integral was discovered in Dominici et al. (Dominici et al. 2012 Proc. R. Soc. A 468, 26672681). Here, we exten...
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an exact integral to sum relation for products of Bessel Functions
arXiv: Classical Analysis and ODEs, 2021Co-Authors: Oliver H E Philcox, Zachary SlepianAbstract:A useful identity relating the infinite sum of two Bessel Functions to their infinite integral was discovered in Dominici et al. (2012). Here, we extend this result to products of $N$ Bessel Functions, and show it can be straightforwardly proven using the Abel-Plana theorem, or the Poisson summation formula. For $N=2$, the proof is much simpler than that of Dominici et al., and significantly enlarges the range of validity.
Sun Kim - One of the best experts on this subject based on the ideXlab platform.
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sums of squares and products of Bessel Functions
Advances in Mathematics, 2018Co-Authors: Bruce C Berndt, Sun Kim, Alexandru Zaharescu, Atul DixitAbstract:Abstract Let r k ( n ) denote the number of representations of the positive integer n as the sum of k squares. We rigorously prove for the first time a Voronoi summation formula for r k ( n ) , k ≥ 2 , proved incorrectly by A.I. Popov and later rediscovered by A.P. Guinand, but without proof and without conditions on the Functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of r k ( n ) and a product of two Bessel Functions, and a series involving r k ( n ) and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G.H. Hardy, and of A.L. Dixon and W.L. Ferrar, as well as of a classical result of A.I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.
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sums of squares and products of Bessel Functions
arXiv: Number Theory, 2017Co-Authors: Bruce C Berndt, Sun Kim, Alexandru Zaharescu, Atul DixitAbstract:Let $r_k(n)$ denote the number of representations of the positive integer $n$ as the sum of $k$ squares. We rigorously prove for the first time a Voronoi summation formula for $r_k(n), k\geq2,$ proved incorrectly by A. I. Popov and later rediscovered by A. P. Guinand, but without proof and without conditions on the Functions associated in the transformation. Using this summation formula we establish a new transformation between a series consisting of $r_k(n)$ and a product of two Bessel Functions, and a series involving $r_k(n)$ and the Gaussian hypergeometric function. This transformation can be considered as a massive generalization of well-known results of G. H. Hardy, and of A. L. Dixon and W. L. Ferrar, as well as of a classical result of A. I. Popov that was completely forgotten. An analytic continuation of this transformation yields further useful results that generalize those obtained earlier by Dixon and Ferrar.
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logarithmic means and double series of Bessel Functions
International Journal of Number Theory, 2015Co-Authors: Bruce C Berndt, Sun KimAbstract:In his lost notebook, Ramanujan recorded two identities involving double series of Bessel Functions that are closely connected with the classical, unsolved circle and divisor problems. In a series of papers with Zaharescu, the authors proved these identities under various interpretations, as well as Riesz mean analogues. In this paper, logarithmic mean analogues, also involving double series of Bessel Functions, are established. Weighted divisor sums involving characters play a central role.
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circle and divisor problems and double series of Bessel Functions
Advances in Mathematics, 2013Co-Authors: Bruce C Berndt, Sun Kim, Alexandru ZaharescuAbstract:Abstract In approximately 1915, Ramanujan recorded two identities involving doubly infinite series of Bessel Functions. The identities were brought to the mathematical public for the first time when his lost notebook was published in 1988, and are connected with the classical, long-standing circle and divisor problems, respectively. We provide a proof of the first identity for the first time by analytically continuing a new kind of Dirichlet series. Delicate estimates of exponential sums are needed, and the new methods we introduce may be of independent interest.
