Mediterr. J. Math. (2018) 15:20 https://doi.org/10.1007/s00009-018-1066-3 1660-5446/18/010001-8 published online January 15, 2018 c  Springer International Publishing AG, part of Springer Nature 2018 An Elementary Function Representation of the Second-Order Moment of the Meyer–K¨ onig and Zeller Operators Ioan Gavrea and Mircea Ivan Abstract. We prove that the second-order moment of the Meyer–K¨onig and Zeller operators is an elementary function and find sharp forms of the related Becker and Nessel inequalities. Mathematics Subject Classification. Primary 41A10, 41A60; Secondary 33C05, 05A19. Keywords. Meyer–K¨onig and Zeller operators, approximation by poly- nomials, asymptotic expansions, hypergeometric functions, combinato- rial identities, Becker–Nessel inequalities. 1. Introduction In this note, we focus on the well-known power series M n (x) := (1 − x) n+1 ∞  k=1  n + k k  k n + k  2 x k , (1.1) where |x| < 1, and n =1, 2,... For x ∈ [0, 1) and n =1, 2,..., the series (1.1) represents, in fact, the second-order moment M n (e 2 ; x) of the classical Meyer–K¨onig and Zeller approximation operators (MKZ for short) [7], M n : C[0, 1] → C[0, 1], in the modified version of Cheney and Sharma [4], M n f (x)= ∞  k=0  n + k k  (1 − x) n+1 x k f  k n + k  , x ∈ [0, 1), M n f (1) = f (1). The second-order central moment of linear operators, in particular, the MKZ second-order central moment, E n (x) := M n (x) − x 2 ,