On the zeros of the hyper-Bessel function

Abstract

In this work, we prove that the zeros of the hyper-Bessel function are located on rays emanating from the origin with the positive real axis as ray, and that the only d-orthogonal Jensen polynomials associated with an entire function in the Laguerre–Pólya class are the d-orthogonal Laguerre polynomials.

Keywords:

• hyper-Bessel function
• d-orthogonal Laguerre polynomials
• Laguerre–Pólya class
• Jensen polynomials

Mathematics Subject Classification:

• 42C05
• 33C45

Acknowledgments

The authors thank Professor Wolfram Koepf for some enlightened maple implementations, and the referee for his/her careful reading of the manuscript and for the suggestion for us.[1]