On the summation of Bessel functions and Hankel transforms (Q1192527)

Important functional relations such as Poisson's summation formula or the functional equation for Riemann's zeta function are equivalent to the well known classic linear transformation formula for Jacobi's elliptic theta function. Along the same lines, the Doetsch-Kober summation formula for Bessel functions has been generalized and transferred by A. Erdelyi to the space of Hankel transforms \[ (H_ \nu f)(s)=\int_ IJ_ \nu(2(st)^{1/2})f(t)dt \quad (s\in\mathbb{R}^ +) \] where the interval \(I\subset\mathbb{R}^ +\), \(H_ \nu\) is a self-adjoint unitary operator on the Hilbert space of functions \(f\in L_ 2(I)\) and \(J_ \nu\) denotes the Bessel function of the first kind of order \(\nu\), \(\text{Re }\nu > - 1/2\). Let \(\Omega=(C,+,.)\) be the vector space of all arithmetic functions \(f:\mathbb{N}\to\mathbb{C}\) and define for \(f,g\in\Omega\), \[ s_ q(n)=\sum_{d|\text{gcd}(n,q)}g(d)h(q/d) \quad (n,q\in\mathbb{N}). \] The \(s_ q(n)\) represents a generalization of Ramanujan's exponential sums \(c_ q(n)\). The author combines arithmetic results on \(s_ q(n)\) with Doetsch- Erdelyi integral transform methods to derive new linear functional relations corresponding to Jacobi's summation formula for Bessel- Schlömilch type series and more generally the series \[ w_ f(q,\nu)=\sum_{n\geq 1}s_ q(n)n^{-\nu}(H_ \nu f)(\pi^ 2n^ 2) \] of Hankel-Erdelyi type. Some important classical formulae for Schlömilch series are included as special cases. The author also derives a general transformation formula of Hankel-Erdelyi type that includes the Bessel-Schlömilch series type.