Abstract
We give explicit expressions for higher order binomial convolutions of sequences of numbers having a finite exponential generating function. Illustrations involving Cauchy, Bernoulli, and Apostol–Euler numbers are presented. In these cases, we obtain formulas easy to compute in terms of Stirling numbers.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington (1964)
Adell, J.A., Lekuona, A.: Binomial convolution and transformations of Appell polynomials. J. Math. Anal. Appl. 456(1), 16–33 (2017). https://doi.org/10.1016/j.jmaa.2017.06.077
Adell, J.A., Lekuona, A.: Closed form expressions for Appell polynomials. Ramanujan J (2018). https://doi.org/10.1007/s11139-018-0026-7
Adell, J.A., Sangüesa, C.: Approximation by \(B\)-spline convolution operators. A probabilistic approach. J. Comput. Appl. Math. 174(1), 79–99 (2005). https://doi.org/10.1016/j.cam.2004.04.001
Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Enlarged Edn. D. Reidel Publishing Co., Dordrecht (1974)
Dilcher, K.: Sums of products of Bernoulli numbers. J. Number Theory 60(1), 23–41 (1996). https://doi.org/10.1006/jnth.1996.0110
Dilcher, K., Vignat, C.: General convolution identities for Bernoulli and Euler polynomials. J. Math. Anal. Appl. 435(2), 1478–1498 (2016). https://doi.org/10.1016/j.jmaa.2015.11.006
Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)
He, Y., Araci, S.: Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials. Adv. Diff. Equ. 2014, 155 (2014). https://doi.org/10.1186/1687-1847-2014-155
Hernández-Llanos, P., Quintana, Y., Urieles, A.: About extensions of generalized Apostol-type polynomials. Results Math. 68(1–2), 203–225 (2015). https://doi.org/10.1007/s00025-014-0430-2
Komatsu, T., Simsek, Y.: Third and higher order convolution identities for Cauchy numbers. Filomat 30(4), 1053–1060 (2016). https://doi.org/10.2298/FIL1604053K
Komatsu, T., Yuan, P.: Hypergeometric Cauchy numbers and polynomials. Acta Math. Hung. 153(2), 382–400 (2017). https://doi.org/10.1007/s10474-017-0744-0
Merlini, D., Sprugnoli, R., Verri, M.C.: The Cauchy numbers. Discrete Math. 306(16), 1906–1920 (2006). https://doi.org/10.1016/j.disc.2006.03.065
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Cambridge University Press, Washington, Cambridge (2010)
Pyo, S.S., Kim, T., Rim, S.H.: Degenerate Cauchy numbers of the third kind. J. Inequal. Appl. 2018, 32 (2018). https://doi.org/10.1186/s13660-018-1626-x
Sun, P.: Product of uniform distribution and Stirling numbers of the first kind. Acta Math. Sin. (Engl. Ser.) 21(6), 1435–1442 (2005). https://doi.org/10.1007/s10114-005-0631-4
Acknowledgements
We would like to thank the referee, whose comments and suggestions greatly improved the final outcome. The authors are partially supported by Research Projects DGA (E-64) and MTM2015-67006-P.
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Adell, J.A., Lekuona, A. Explicit Expressions for Higher Order Binomial Convolutions of Numerical Sequences. Results Math 74, 80 (2019). https://doi.org/10.1007/s00025-019-1005-z
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DOI: https://doi.org/10.1007/s00025-019-1005-z