Closed-form expressions for moments of a class of beta generalized distributions

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The logistic-X family of distributions and its applications (2016, background): “Cordeiro and Nadarajah (2011) determined τ (n, a) for some well-known distributions such as normal, beta, gamma, and Weibull distributions.”
The Exponentiated Half-Logistic Family of Distributions: Properties and Applications (2014, methodology): “…the EHLN cumulative function (18) (with 𝜇 = 0 and 𝜎 = 1 ) as From the series expansion for the error function erf (⋅) we obtain a series expansion from (20) (with 𝜇 = 0 and 𝜎 = 1 ) given by The EHLN moments can be obtained from (36) and the PWMs 𝜏 𝑟,𝑘 given by Cordeiro and Nadarajah [14].”
A New Weibull-G Family of Distributions (2015, unknown): “11 Following similar algebraic developments of Nadarajah et al. [22], we can write fi:n(x) = ∞∑ r,k=0 mr,k hr+k+1(x),(6.1) where hr+k+1(x) is the exp-G density function with power parameter r + k + 1, mr,k = n!”
The Exponentiated Generalized Class of Distributions (2021, background): “They are given by (Cordeiro and Nadarajah, 2011) τr,j = a −jB(a, b)−(j+1)B(b, r + a(j + 1))F 1:2 1:1 ((r + a(j + 1)) : (1− b, a); · · · ; (1− b, a) : (b+ r + a(j + 1)) : (a+ 1); · · · ; (a+ 1); 1, · · · , 1), where the generalized Kampé de Fériet function is defined by F C:D ((a) : (b1); · · · ; (bn”