On the properties of zeros of Bessel series in the real line

ABSTRACT

For a given sequence of real numbers $\mathfrak{a}=(a_m{)}_{m\ge 1}$, we define the function $U_{\mu ,\nu }^{\mathfrak{a}}\left(x\right)=\frac{2^{\mu }\mathrm{\Gamma }(\mu +1)}{x^{\mu }}\sum _{m\ge 1}\frac{a_m}{j_{m,\nu }^{\mu }}{J}_{\mu }\left(j_{m,\nu }x\right),x\in (0,+\mathrm{\infty }),$ where $\mu ,\nu >-1$, $J_{\mu }$ denotes the Bessel function of order μ, and $(j_{m,\nu }{)}_{m\ge 1}$ are the positive zeros of $J_{\nu }$. In this paper, we study the function $U_{\mu ,\nu }^{\mathfrak{a}}$ and some outstanding instances of it on the whole real line and propose a number of conjectures about its positive zeros.

Keywords:

• Bessel functions
• zeros
• Bessel series

AMS CLASSIFICATIONS:

• Primary 33C10
• Secondary 42C10
• 34C10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the spanish Ministerio de Ciencia, Innovación y Universidades under Grants PGC2018-096504-B-C31 and PGC2018-096504-B-C32; Junta de Andalucía under Grants FQM-262 and US-1254600; Gobierno de Aragón under Grant E26_17R; and Feder Funds of the European Union.