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AN ASYMPTOTIC EXPANSION FOR THE FIRST DERIVATIVE OF THE HURWITZ-TYPE EULER ZETA FUNCTION

  • MIN-SOO KIM (Department of Mathematics Education, Kyungnam University)
  • Received : 2023.08.30
  • Accepted : 2023.11.06
  • Published : 2023.11.30

Abstract

The Hurwitz-type Euler zeta function ζE(z, q) is defined by the series ${\zeta}_E(z,\,q)\,=\,\sum\limits_{n=0}^{\infty}{\frac{(-1)^n}{(n\,+\,q)^z}},$ for Re(z) > 0 and q ≠ 0, -1, -2, . . . , and it can be analytic continued to the whole complex plane. An asymptotic expansion for ζ'E(-m, q) has been proved based on the calculation of Hermite's integral representation for ζE(z, q).

Keywords

Acknowledgement

This work was supported by the Kyungnam University Foundation Grant, 2022.

References

  1. M. Abramowitz and I. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover, New York, 1972.
  2. V.S. Adamchik, A class of logarithmic integrals, In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, Kihei, HI, ACM, New York, 1997, 1-8.
  3. V.S. Adamchik, Polygamma functions of negative order, J. Comput. Appl. Math. 100 (1998), 191-199. https://doi.org/10.1016/S0377-0427(98)00192-7
  4. V.S. Adamchik, The multiple gamma function and its application to computation of series, Ramanujan J. 9 (2005), 271-288. https://doi.org/10.1007/s11139-005-1868-3
  5. V.S. Adamchik, On the Hurwitz function for rational arguments, Appl. Math. Comput. 187 (2007), 3-12. https://doi.org/10.1016/j.amc.2006.08.096
  6. T.M. Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.
  7. J. Choi, Y.J. Cho and H.M. Srivastava, Series involving the zeta function and multiple gamma functions, Appl. Math. Comput. 159 (2004), 509-537.
  8. J. Choi, H.M. Srivastava, A family of log-gamma integrals and associated results, J. Math. Anal. Appl. 303 (2005), 436-449. https://doi.org/10.1016/j.jmaa.2004.08.043
  9. J. Choi and H.M. Srivastava, The multiple Hurwitz zeta function and the multiple Hurwitz-Euler eta function, Taiwanese J. Math. 15 (2011), 501-522.
  10. E. Elizalde, An asymptotic expansion for the first derivative of the generalized Riemann zeta function, Math. Comp. 47 (1986), 347-350. https://doi.org/10.1090/S0025-5718-1986-0842140-X
  11. E. Elizalde and A. Romeo, An integral involving the generalized zeta function, Int. J. Math. Math. Sci. 13 (1990), 453-460. https://doi.org/10.1155/S0161171290000679
  12. S. Hu and M.-S. Kim, Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives, Preprint, 2023. https://arxiv.org/abs/2103.15528.
  13. M.-S. Kim, Some series involving the Euler zeta function, Turkish J. Math. 42 (2018), 1166-1179. https://doi.org/10.3906/mat-1704-107
  14. C.S. Ryoo, On the (p, q)-analogue of Euler zeta function, J. Appl. Math. Inform. 35 (2017), 303-311. https://doi.org/10.14317/jami.2017.303
  15. R. Seri, A non-recursive formula for the higher derivatives of the Hurwitz zeta function, J. Math. Anal. Appl. 424 (2015), 826-834. https://doi.org/10.1016/j.jmaa.2014.08.012
  16. Z.-W. Sun, Introduction to Bernoulli and Euler polynomials, A Lecture Given in Taiwan on June 6, 2002. http://maths.nju.edu.cn/~zwsun/BerE.pdf.
  17. K.S. Williams and N.Y. Zhang, Special values of the Lerch zeta function and the evaluation of certain integrals, Proc. Amer. Math. Soc. 119 (1993), 35-49. https://doi.org/10.1090/S0002-9939-1993-1172963-7