We study the asymptotic behaviour of bessel functions associated to root systems of type An-1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit...
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We study the asymptotic behaviour of bessel functions associated to root systems of type An-1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik-Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of beta-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F = R, C, H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits. (c) 2024 The Author(s). Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). This is an open access article under the CC BY license (http://***/licenses/by/4.0/).
In this paper, on the complex field C, we prove two integral formulae for the Hankel-Mellin transform and the double Fourier-Mellin transform of bessel functions, both resulting the hypergeometric function. As two app...
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In this paper, on the complex field C, we prove two integral formulae for the Hankel-Mellin transform and the double Fourier-Mellin transform of bessel functions, both resulting the hypergeometric function. As two applications, we use the former integral formula to make explicit the spectral formula of Bruggeman and Motohashi for the fourth moment of Dedekind zeta function over the Gaussian number field Q( i ) and to establish a spectral formula for the Heckeeigenvalue twisted second moment of central L- values for the Picard group PGL2(Z[i]). Moreover, we develop the theory of distributional Hankel transform on C N {0 }. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The purpose of this article is to verify the conjectures of the previous paper in the particular case of GLp4q. We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interc...
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A method to efficiently evaluate integrals containing the product of three bessel functions of the first kind and of any non negative real order is presented. The numerical problem is particularly challenging since st...
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A method to efficiently evaluate integrals containing the product of three bessel functions of the first kind and of any non negative real order is presented. The numerical problem is particularly challenging since standard integration techniques are completely unsuccessful in integrating such anomalously oscillating (and possibly slow decaying) functions. The proposed method is based on a decomposition of such a product into a sum of functions which asymptotically approach sinusoidal functions, for which integration schemes based on integration then summation procedures followed by extrapolation methods can be applied. Different extrapolation procedures are compared in order to identify the most efficient extrapolation strategy. Several numerical examples and comparisons with known analytic results are provided to show the robustness and the accuracy of the proposed approach and to help in identifying the most efficient and reliable extrapolation scheme. Particular attention is given to a class of integrals emerging in many fields of physics, in particular in quantum mechanics and particle physics, showing that the proposed method can be even more efficient (sometimes hundred of times faster) than the available analytical formulas.
Series involving hypergeometric functions are used to derive, extend and evaluate integrals involving the product of two bessel functions of the first kind Ju(az) Jv(bz) with order u, v, studied by Landau et al. The m...
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A representation of solutions of the one-dimensional Dirac equation is obtained. The solutions are represented as Neumann series of bessel functions. The representations are shown to be uniformly convergent with respe...
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Integrals involving highly oscillatory bessel functions are notoriously challenging to compute using conventional integration techniques. While several methods are available, they predominantly cater to integrals with...
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Monotonicity with respect to the order v of the magnitude of general bessel functions C-v(x) = aJ(v)(x) + bY(v)(x) at positive stationary points of associated functions is derived. In particular, the magnitude of C-v ...
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Monotonicity with respect to the order v of the magnitude of general bessel functions C-v(x) = aJ(v)(x) + bY(v)(x) at positive stationary points of associated functions is derived. In particular, the magnitude of C-v at its positive stationary points is strictly decreasing in v for all positive v. It follows that sup(x)\J(v)(x)\ strictly decreases from 1 to 0 as v increases from 0 to infinity. The magnitude of x(1/2)C(v)(x) at its positive stationary points is strictly increasing in v. It follows that sup(x)\x(1/2)J(v)(x)\ equals root 2/pi for 0 less than or equal to v less than or equal to 1/2 and strictly increases to co as v increases from 1/2 to infinity. It is shown that v(1/8) sup(x) \J(v)(x)\ strictly increases from 0 to b = 0.674885... as v increases from 0 to infinity. Hence for all positive v and real x, \J(v)(x)\ < bv(-1/3) where b is the best possible such constant. Furthermore, for all positive v and real x, \J(v)(x)\ less than or equal to c\x\(-1/3) where c = 0.7857468704... is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.
In the context of the Kuznetsov trace formula, we outline the theory of the bessel functions on GL ( n ) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ra...
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In the context of the Kuznetsov trace formula, we outline the theory of the bessel functions on GL ( n ) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element;as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n ) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL (4) bessel functions. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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