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Register a new userSymbolic methods for the evaluation of sum rules of Bessel functions
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The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics.
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In this note, we derive the closed-form expression for the summation of series $sum_{n=0}^{infty}nJ_n(x)partial J_n/partial n$, which is found in the calculation of entanglement entropy in 2-d bosonic free field, in terms of $Y_0$, $J_0$ and an integral involving these two Bessel functions. Further, we point out the integral can be expressed as a Meijer G function.
We solve the Cauchy problem for the $n$-dimensional wave equation using elementary properties of the Bessel functions.
We revisit two classical formulas for the Bessel function of the first kind, due to von Lommel and Weber-Schafheitlin, in a probabilistic setting. The von Lommel formula exhibits a family of solutions to the van Dantzig problem involving the generalized semi-circular distributions and the first hitting times of a Bessel process with positive parameter, whereas the Weber-Schafheitlin formula allows one to construct non-trivial moments of Gamma type having a signed spectral measure. Along the way, we observe that the Weber-Schafheitlin formula is a simple consequence of the von Lommel formula, the Fresnel integral and the Selberg integral.
We review and further develop the theory of $E$-orbit functions. They are functions on the Euclidean space $E_n$ obtained from the multivariate exponential function by symmetrization by means of an even part $W_{e}$ of a Weyl group $W$, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group $W$. The $E$-orbit functions, determined by integral parameters, are invariant with respect to even part $W^{rm aff}_{e}$ of the affine Weyl group corresponding to $W$. The $E$-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain $F^{e}$ of the group $W^{rm aff}_{e}$ (the discrete $E$-orbit function transform).
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
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