Selecting the correct multivariate Stieltjes representation

Determine, for multivariate functions arising in quantum field theory such as multivariate Feynman integrals, which of the two multivariate Stieltjes integral representations—Version 1 with separated factors ∫ μ(t_1,…,t_n)/∏_i(t_i+z_i) or Version 2 with a linear denominator ∫ μ(t_1,…,t_n)/(1+∑_i t_i z_i)—is applicable in general, and develop criteria that predict the appropriate representation for more complicated multivariate integrals.

Background

Two natural generalizations of the Stieltjes representation to several variables are discussed: (i) a product of one‑variable Stieltjes kernels and (ii) a single linear denominator in the variables. Some physics examples admit both forms, while others (e.g., certain Mandelstam representations) manifestly fit only one.

A general understanding of when each representation applies is lacking; resolving this would guide multivariate Padé approximations and positivity analyses.