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CLOSED FORMS FOR DEGENERATE BERNOULLI POLYNOMIALS
Published online by Cambridge University Press: 10 January 2020
Abstract
Qi and Chapman [‘Two closed forms for the Bernoulli polynomials’, J. Number Theory159 (2016), 89–100] gave a closed form expression for the Bernoulli polynomials as polynomials with coefficients involving Stirling numbers of the second kind. We extend the formula to the degenerate Bernoulli polynomials, replacing the Stirling numbers by degenerate Stirling numbers of the second kind.
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- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
The work is supported by the National Natural Science Foundation of China (Grant No. 11671197).
References
Adelberg, A., ‘A finite difference approach to degenerate Bernoulli and Stirling polynomials’, Discrete Math. 140 (1995), 1–21.CrossRefGoogle Scholar
Carlitz, L., ‘A degenerate Staudt–Clausen theorem’, Arch. Math. (Basel) 7 (1956), 28–33.CrossRefGoogle Scholar
Carlitz, L., ‘Degenerate Stirling, Bernoulli and Eulerian numbers’, Util. Math. 15 (1979), 51–88.Google Scholar
Comtet, L., Advanced Combinatorics: The Art of Finite and Infinite Expansions, revised and enlarged edn (D. Reidel, Dordrecht, 1974).CrossRefGoogle Scholar
Howard, F. T., ‘Bell polynomials and degenerate Stirling numbers’, Rend. Semin. Mat. Univ. Padova 61 (1979), 203–219.Google Scholar
Howard, F. T., ‘Degenerate weighted Stirling numbers’, Discrete Math. 57 (1985), 45–58.CrossRefGoogle Scholar
Howard, F. T., ‘Explicit formulas for degenerate Bernoulli numbers’, Discrete Math. 162 (1996), 175–185.CrossRefGoogle Scholar
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, 2nd edn, Graduate Texts in Mathematics, 84 (Springer, New York, 1990).CrossRefGoogle Scholar
Kim, T. and Kim, D. S., ‘Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind’, Sci. China Math. 62 (2019), 999–1028.CrossRefGoogle Scholar
Kim, T., Kim, D. S. and Kwon, H.-I., ‘Some identities relating to degenerate Bernoulli polynomials’, Filomat 30 (2016), 905–912.CrossRefGoogle Scholar
Murty, H. R., Introduction to p-adic Analytic Number Theory, AMS/IP Studies in Advanced Mathematics, 27 (American Mathematical Society; International Press, Providence, RI; Somerville, MA, 2002).Google Scholar
Qi, F. and Chapman, R. J., ‘Two closed forms for the Bernoulli polynomials’, J. Number Theory 159 (2016), 89–100.CrossRefGoogle Scholar
Washington, L. C., Introduction to Cyclotomic Fields, 2nd edn, Graduate Texts in Mathematics, 83 (Springer, New York, 1997).CrossRefGoogle Scholar
Young, P. T., ‘Congruences for degenerate number sequences’, Discrete Math. 270 (2003), 279–289.CrossRefGoogle Scholar
Young, P. T., ‘Degenerate Bernoulli polynomials, generalized factorial sums, and their applications’, J. Number Theory 128 (2008), 738–758.CrossRefGoogle Scholar
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