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This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$q(\\tau ,w)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>\u03c4<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>w<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> have already been addressed in the scientific literature. The contribution of this paper is twofold. First, we place these methods in the classical framework of Krylov-Lanczos (polynomial-rational) techniques for accelerating Fourier series. This allows us to apply the convergence results developed in this context to our function. Second, we design a new acceleration scheme. Some numerical results are presented to show the effectiveness of the proposed algorithms.<\/jats:p>","DOI":"10.1007\/s10444-025-10231-1","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T08:10:23Z","timestamp":1744272623000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computing the action of the matrix generating function of Bernoulli polynomials on a vector with an application to non-local boundary value problems"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4537-2444","authenticated-orcid":false,"given":"Lidia","family":"Aceto","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8000-4906","authenticated-orcid":false,"given":"Luca","family":"Gemignani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"10231_CR1","doi-asserted-by":"crossref","unstructured":"Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. 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