eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2018-06-16 6 3 90 92 10.12691/tjant-6-3-4 TJANT2018634 article Anthony G. Shannon tshannon38@gmail.com 1 Ömür Deveci 2 Emeritus Professor, University of Technology Sydney, NSW 2007, Australia, Fellow, Warrane College, University of New South Wales, Kensington NSW 2033 Department of Mathematics, Faculty of Science & Letters, Kafkas University, 36100, Kars, Turkey This paper considers generalizations of Bernoulli and Euler numbers to clarify and extend some known relations studied by Morgan Ward. It does this with the Euler-Maclaurin sum formula. It relates the mappings to category theory as a means of applying the ideas further. http://pubs.sciepub.com/tjant/6/3/4/tjant-6-3-4.pdf Special function Bernoulli polynomial Euler function difference operator normal sequence divisibility sequence difference operator generalized integer Fermatian numbers Euler-Maclaurin sum-formula commutative diagram category