## 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.

### MSC:

 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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