Many algebraic equations satisfied by the zeros of the classical polynomials and of the Bessel functions are reported. Some of them are collected from recent papers; several of them are new; most of them display remarkable diophantine features. Certain matrices constructed with arbitrary numbers rather than the zeros of special functions, but displaying analogous diophantine properties, are also exhibited.

Properties of the zeros of the classical polynomials and of the Bbessel functions

Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d \le a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d >0$. Various other characterizations of the distribution of $M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $M_d$ is described both as $d$ tends to infinity and as $d$ tends to zero.

/pdf/the-law-of-the-maximum-of-a-bessel-bridge-w6novsqs90.pdf

The Law of the Maximum of a Bessel Bridge

Spectral properties of the biconfluent Heun differential equation

S. Ahmed

Papers

On the zeros of Bessel functions. - IV

On the zeros of Bessel functions.- III

On the zeros of Bessel functions. - V