Invariance of the numbers of positive and negative zeros under φ for hypergeometric B_φ

Establish whether, for parameters φ = (φ1,...,φN) with φi > 0 and for any A in the normalized Laguerre–Pólya class, the numbers of positive zeros n+(φ) and negative zeros n−(φ) of the n-th Brenke polynomial pn(x) generated by A and associated to the hypergeometric function B_φ(z) = 0F_N(-; φ1,...,φN; z) are independent of φ.

Background

Section 6 shows that B_φ(z) = 0F_N(-; φ1,...,φN; z) with φi > 0 belongs to the JPS class, so for any A ∈ L-P0 the associated Brenke polynomials have only real zeros with strong simplicity and interlacing properties (Theorem 6.3).

Beyond these results, the authors raise parameter-dependent questions about how the zeros behave as the hypergeometric parameters φ vary, beginning with whether the counts of positive and negative zeros are invariant under φ.