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

Durán, Antonio J.; Pérez, Mario; Varona, Juan L.

Proc. Am. Math. Soc. 150, No. 2, 763-778 (2022).

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

Keywords:

Bessel functions; Bessel series; Fourier series; Appell polynomials; Bernoulli polynomials; hypergeometric functions

Software:

DLMF

References:

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