A method for summing Bessel series and a couple of illustrative examples. (English) Zbl 1497.33007

Summary: For \(\mu,\nu>-1\), we consider the Bessel series \[ U_{\mu,\nu}^{\mathfrak{a}}(x)=\frac{2^\mu\Gamma(\mu+1)}{x^\mu} \sum_{m\geq 1}\frac{a_m}{j_{m,\nu}^{\mu+1/2}} J_\mu (j_{m,\nu}x), \] where \((j_{m,\nu})_{m\geq 1}\) are the positive zeros of \(J_\nu\) and \(\mathfrak{a}=(a_m)_{m\geq 1}\) is a sequence of real numbers satisfying \(\sum_{m\geq 1}|a_m|/j_{m,\nu}^{\mu +1/2}<+\infty\). 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.


33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
11B68 Bernoulli and Euler numbers and polynomials
33C20 Generalized hypergeometric series, \({}_pF_q\)
42A10 Trigonometric approximation


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