Fuzzy Optimization and Decision Making https://doi.org/10.1007/s10700-020-09338-5 A relation between moments of Liu process and Bernoulli numbers Guanzhong Ma 1 · Xiangfeng Yang 2 · Xiao Yao 3 Accepted: 7 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract This paper finds a relation between moments of Liu process and Bernoulli numbers. Firstly, by an exponential generating function of Bernoulli numbers, a useful integral formula is obtained. Secondly, based on this integral formula, the moments of a normal uncertain variable and Liu process are expressed via Bernoulli numbers. Keywords Uncertainty theory · Liu process · Bernoulli numbers · Inverse uncertainty distribution 1 Introduction In order to handle the belief degree of humans, uncertainty theory was founded by Liu (2007, 2009) on the foundation of normality, duality, subadditivity, and product axioms. Afterward, this theory was improved by Liu (2010) and Liu (2015). A series of fundamental concepts in uncertainty theory, such as uncertain variable, expectation, variance, and moment, were proposed by Liu (2007). For calculating the expectation of a function of uncertain variables, a compelling method was discovered by Liu and Ha (2010). Based on the inverse uncertainty distribution, Yao (2015) introduced a new approach to compute the variance. Later on, Sheng and Kar (2015) extended Yao’s result to the moments of the uncertain variable. These formulae, in Liu and Ha (2010), B Guanzhong Ma maguanzhong75@aynu.edu.cn Xiangfeng Yang yangxf@uibe.edu.cn Xiao Yao yaoxiao@nankai.edu.cn 1 School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China 2 School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China 3 School of Mathematics Sciences, Nankai University, Tianjin 300071, China 123