Monotonicity of zeros with respect to φ for hypergeometric B_φ

Prove that for parameters φ, ψ ∈ (0, ∞)^N with φ ≤ ψ componentwise and for A in the normalized Laguerre–Pólya class, the j-th positive zero ζ_j^+(φ) of the n-th Brenke polynomial pn(x) generated by A and associated to B_φ(z) = 0F_N(-; φ1,...,φN; z) is nondecreasing in φ (i.e., ζ_j^+(φ) ≤ ζ_j^+(ψ)), while the j-th negative zero ζ_j^−(φ) is nonincreasing in φ (i.e., ζ_j^−(φ) ≥ ζ_j^−(ψ)).

Background

After proving simplicity and interlacing for fixed φ (Theorem 6.3), the authors turn to parameter sensitivity. They define the componentwise order φ ≤ ψ and formulate a conjectural monotonicity of individual zeros of pn(x) as functions of φ.

Such monotonicity would provide a refined understanding of the dependence of zero locations on the hypergeometric parameters beyond mere interlacing in degree.