A method for summing Bessel series and a couple of illustrative examples

Abstract:

For $\mu ,\nu >-1$, we consider the Bessel series \[ U_{\mu ,\nu }^{\mathfrak {a}}(x) = \frac {2^\mu \Gamma (\mu +1)}{x^\mu } \sum _{m\ge 1} \frac {a_m}{j_{m,\nu }^{\mu +1/2}} J_\mu (j_{m,\nu } x), \] where $(j_{m,\nu })_{m\ge 1}$ are the positive zeros of $J_\nu$ and $\mathfrak {a} = (a_m)_{m\ge 1}$ is a sequence of real numbers satisfying $\sum _{m\ge 1} {|a_m|}/{j_{m,\nu }^{\mu +1/2}} < +\infty$. We propose a method for computing in a closed form the sum of the Bessel series $U_{\mu ,\nu }^{\mathfrak {a}}$ assuming that for a particular value $\eta$ of the parameter $\mu$ a closed expression for $U_{\eta ,\nu }^{\mathfrak {a}}$ as a power series of $x$ (not necessarily with integer exponents) is known. We illustrate the method with some examples. One of them is related to the sine coefficients of the function $1-x^s$, $s>-1$. The closed form of the sum is then given in terms of a generalization of the Bernoulli numbers.