Abstract:A large class of Appell polynomial sequences {p n (x)} ∞ n=0 are special values at the negative integers of an entire function F (s, x), given by the Mellin transform of the generating function for the sequence. For the Bernoulli and Apostol-Bernoulli polynomials, these are basically the Hurwitz zeta function and the Lerch transcendent. Each of these have well-known Fourier series which are proved in the literature using various techniques. Here we find the latter Fourier series by directly calculating the coe… Show more
“…where Pn (x) = P n (x) as in the previous section for n ≥ 1, and P 0 (x) = 0. Based on [10], we may introduce a new variable z and observe (3.2) is equivalent to…”
We study the continuity properties of a generalized Davenport Fourier expansion we recently discovered, by imposing conditions on the coefficients. We also put our expansion into perspective from the position of Appell sequences.
“…where Pn (x) = P n (x) as in the previous section for n ≥ 1, and P 0 (x) = 0. Based on [10], we may introduce a new variable z and observe (3.2) is equivalent to…”
We study the continuity properties of a generalized Davenport Fourier expansion we recently discovered, by imposing conditions on the coefficients. We also put our expansion into perspective from the position of Appell sequences.
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