Abstract
Improving an old idea of Hermite, we associate to each natural number k a modified zeta function of order k. The evaluation of the values of these functions F k at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta values.
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References
Arakawa, T., Kaneko, M.: Multiple zeta values, Poly–Bernoulli numbers and related zeta functions. Nagoya Math. J. 153, 189–209 (1999)
Boyadzhiev, K.: Harmonic number identities via Euler’s transform. J. Integer Seq. 12, Article 09.6.1 (2009)
Candelpergher, B., Coppo, M.A., Delabaere, E.: La sommation de Ramanujan. Enseign. Math. 43, 93–132 (1997)
Candelpergher, B., Gadiyar, H., Padma, R.: Ramanujan summation and the exponential generating function \(\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)\). Ramanujan J. 21, 99–122 (2010)
Chen, X., Chu, W.: Dixon’s F 2(1)-series and identities involving harmonic numbers and the Riemann zeta function. Discrete Math. 310, 83–91 (2010)
Choi, J., Srivastava, H.M.: Explicit evaluation of Euler and related sums. Ramanujan J. 10, 51–70 (2005)
Coppo, M.-A.: Nouvelles expressions des formules de Hasse et de Hermite pour la fonction zêta d’Hurwitz. Expo. Math. 27, 79–86 (2009)
Coppo, M.-A., Candelpergher, B.: The Arakawa–Kaneko zeta function. Ramanujan J. 22, 153–162 (2010)
Dilcher, K.: Some q-series identities related to divisors functions. Discrete Math. 145, 83–93 (1995)
Flajolet, P., Sedgewick, R.: Mellin transforms and asymptotics: finite differences and Rice’s integrals. Theor. Comput. Sci. 144, 101–124 (1995)
Hermite, C.: Extrait de quelques lettres de M.Ch. Hermite à M.S. Pincherle. Ann. Mat. Pura Appl. 5, 55–72 (1901)
Merlini, D., Sprugnoli, R., Verri, C.: The Cauchy numbers. Discrete Math. 306, 1906–1920 (2006)
Schiff, J.: The Laplace Transform: Theory and Applications. Springer, New York (1999)
Zeidler, E.: Quantum Field Theory I: Basics in Mathematics and Physics. Springer, Berlin (2006)
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Candelpergher, B., Coppo, MA. A new class of identities involving Cauchy numbers, harmonic numbers and zeta values. Ramanujan J 27, 305–328 (2012). https://doi.org/10.1007/s11139-011-9361-7
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DOI: https://doi.org/10.1007/s11139-011-9361-7
Keywords
- Cauchy numbers
- Bell polynomials
- Harmonic numbers
- Laplace–Borel transform
- Mellin transform
- Zeta values
- Ramanujan summation
- Hermite’s formula