Generalized series of Bessel functions

Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals

“…or in more perfect analogy with ( The beautiful formula (6) was in fact established earlier by Al-Jarrah, Dempsey, and Glasser [2] (compare also with Theorem 9 in [5]) by a very different method. Note however that these formulas do not seem to have been noticed previously in the vast classical literature on Bessel functions.…”

On New Formulas for the Cumulative Distribution Function of the Noncentral Chi-Square Distribution

“…The problem of summing the Neumann series (2.1) has been widely considered in mathematical literature [1,3,9,14,18,23,27] and concerning the result derived in the previous theorem we can conclude that the problem of deriving a new formula for F 2n,a is equivalent to a problem of deriving a new closed-form expression for the Neumann series given in (2.2).…”

Green's functions on finite lattices and their connection to the infinite lattice limit

“…For simplicity we also confine the energy parameter l to the region outside the scattering states, l ! 2: Let us consider a simple generalization of the formula expressing (1) in terms of modified Bessel functions [7][8][9][10] G sq X, Y ð Þ¼…”

“…The structure of this exact expression is surprisingly simple and indicates the possibility of extension to other lattices. In particular, (7) contains the periodic symmetry of the LGF represented by the transformation X → (N − X) and Y → (N − Y ) . For X, Y << N and γ = 1 we find in the thermodynamic limit, N → ∞ :…”

“…The further development of the approach was based on the proof of several exact expressions for the finite lattice sums and the discrete Fourier transform in one dimension [6]. We have recently become aware of the paper [7] where some similar exact formulas have been derived and note that our paper in [6] was submitted before publication of [7].…”