Sanchez-Ortiz, Jorge Generalized fractional-order Bernoulli functions via Riemann-Liouville operator and their applications in the evaluation of Dirichlet series. (English) Zbl 1470.05019 Abstr. Appl. Anal. 2018, Article ID 4875916, 5 p. (2018). Summary: In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series. MSC: 05A15 Exact enumeration problems, generating functions 11B68 Bernoulli and Euler numbers and polynomials 26A33 Fractional derivatives and integrals 33E12 Mittag-Leffler functions and generalizations PDF BibTeX XML Cite \textit{J. Sanchez-Ortiz}, Abstr. Appl. Anal. 2018, Article ID 4875916, 5 p. (2018; Zbl 1470.05019) Full Text: DOI References: [1] Euler, L., Methodus generalis summandi progressiones, Commentarii Academiae Scientiarum Petropolitanae, 6, 68-97 (1738) [2] Lehmer, D. H., A new approach to Bernoulli polynomials, The American Mathematical Monthly, 95, 10, 905-911 (1988) · Zbl 0663.10009 [3] Natalini, P.; Bernardini, A., A generalization of the bernoulli polynomials, Journal of Applied Mathematics, 3, 155-163 (2003) · Zbl 1019.33011 [4] Balanzario, E. P., A generalized Euler-Maclaurin formula for the Hurwitz zeta function, Mathematica Slovaca, 56, 3, 307-316 (2006) · Zbl 1141.11041 [5] Balanzario, E. P.; Sanchez-Ortiz, J., A generating function for a class of generalized Bernoulli polynomials, Ramanujan Journal, 19, 1, 9-18 (2009) · Zbl 1195.11028 [6] Rahimkhani, P.; Ordokhani, Y.; Babolian, E., Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations, Applied Numerical Mathematics, 122, 66-81 (2017) · Zbl 1375.65175 [7] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), San Diego, Calif, USA: Elsevier, San Diego, Calif, USA · Zbl 1092.45003 [8] Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S. V., Mittag-Leffler Functions, Related Topics and Applications (2014), Berlin, Germany: Springer, Berlin, Germany · Zbl 1309.33001 [9] Agnew, R. P., Differential Equations (1960), New York, NY, USA: McGraw-Hill, New York, NY, USA [10] Balanzario, E. P., Evaluation of Dirichlet series, The American Mathematical Monthly, 108, 10, 969-971 (2001) · Zbl 1025.11027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.