Connections of the Poisson weight function to overdispersion and underdispersion

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Integer-valued autoregressive models for counts showing underdispersion (2013, background): “the above-mentioned COM distribution with a weight function not depending on λ [20].”
Discrete associated kernels method and extensions (2011, methodology): “[14], and references therein) for that we can use the adapted bandwidth h0 = h0(n;K, f) of h as solution of n ∑”
Underdispersion models: Models that are “under the radar” (2017, methodology): “(2006) consider modeling the data via a Poisson and a GC regression, respectively, while Lord and Guikema (2012) further consider a CMP model.”
Discrete dispersion models and their Tweedie asymptotics (2014, methodology): “Turning now to the question of infinite divisibility, we note that Kokonendji et al. (2008) argued that the COM-Poisson is a weighted Poisson distribution of the form (2.20) with a logconvex (logconcave) weight function for 0 < ν < 1 (ν > 1), and is hence overdispersed (underdispersed) with respect…”