On an open problem of Chen and Mortici concerning the Euler–Mascheroni constant

Chen and Mortici [Ch.-P. Chen, C. Mortici, New sequence converging towards the Euler–Mascheroni constant, Comput. Math. Appl. (2011); http://dx.doi.org/10.1016/j.camwa.2011.03.099] proposed an open problem: for a given positive integer s , find the constants a i ( i = 0 , 1 , 2 , … , s ) such that ∑ k = 1 n 1 k − log ( n + ∑ i = 0 s a i n − i ) is the fastest sequence which would converge to the Euler–Mascheroni constant γ . Using logarithmic type Bell polynomials, we solve this problem. The main result shows that the a i ’s can be recursively determined.