Abstract
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.
Highlights
We explore here some formal aspects of generalized Bernoulli numbers in terms of normal divisibility sequences as defined by Morgan Ward [1,2]
Is an analogous Euler function and {un} is a sequence which is normal in that u(s,t) = and divisible in that us | ut whenever s | t
Since B2n+1 = 0, n ≥ 1, when {un} = Z, these reduce to the ordinary Bernoulli numbers [10]
Summary
We explore here some formal aspects of generalized Bernoulli numbers in terms of normal divisibility sequences as defined by Morgan Ward [1,2]. The generalized Bernoulli polynomials in question have the form Bn (x) such that. Is an analogous Euler function and {un} is a sequence which is normal in that u(s,t) = (us ,ut ) and divisible in that us | ut whenever s | t. 1. Bn (0) = Bn , a generalized Bernoulli number. Ward’s generalized coefficients were rediscoveries of work by Fontené [3]. That on equating coefficients of t, we get the rather neat result that extends Ward [4]
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