Acknowledgement
The author would like to thank Prof. Su Hu for his helpful comments and suggestions.
References
- M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
- A.T. Benjamin, J. Lentfer and T.C. Martinez, Counting on Euler and Bernoulli number identities, Fibonacci Quart. 58 (2020), 30-33.
- A.T. Benjamin and J.J. Quinn, An alternate approach to alternating sums: a method to DIE for, College Math. J. 39 (2008), 191-201. https://doi.org/10.1080/07468342.2008.11922293
- K.-W. Chen, Congruences for Euler numbers, Fibonacci Quart. 42 (2004), 128-140.
- J. Choi and H.M. Srivastava, The multiple Hurwitz zeta function and the multiple HurwitzEuler eta function, Taiwanese J. Math. 15 (2011), 501-522. https://doi.org/10.11650/twjm/1500406218
- M. Eie and Y.L. Ong, A generalization of Kummer's congruences, Abh. Math. Sem. Univ. Hamburg 67 (1997), 149-157. https://doi.org/10.1007/BF02940825
- R.C. Entringer, A Combinatorial Interpretation of the Euler and Bernoulli Numbers, Nieuw Arch. Wiskd. 14 (1966), 241-246.
- Y. He, and Q. Liao, Some congruences involving Euler numbers, Fibonacci Quart. 46/47 (2008), 225-234.
- N.M. Katz, p-adic L-functions via moduli of elliptic curves, Algebraic geometry, Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974, pp. 479-506, Amer. Math. Soc., Providence, R. I., 1975.
- T. Kim, On the analogs of Euler numbers and polynomials associated with p-adic q-integral on Zp at q = -1, J. Math. Anal. Appl. 331 (2007), 779-792. https://doi.org/10.1016/j.jmaa.2006.09.027
- M.-S. Kim and J.-W. Son, On a multidimensional Volkenborn integral and higher order Bernoulli numbers, Bull. Austral. Math. Soc. 65 (2002), 59-71. https://doi.org/10.1017/S0004972700020062
- M.-S. Kim and S. Hu, On p-adic Hurwitz-type Euler zeta functions, J. Number Theory 132 (2012), 2977-3015. https://doi.org/10.1016/j.jnt.2012.05.037
- N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Second edition, Graduate Texts in Mathematics, 58, Springer-Verlag, New York, 1984.
- T. Kubota und H.W. Leopoldt, Eine p-adische Theorie der Zetawerte. I. Einfuhrung der p-adischen Dirichletschen L-Funktionen, J. Reine Angew. Math. 214/215 (1964), 328-339. https://doi.org/10.1515/crll.1964.214-215.328
- E.E. Kummer, uber eine allgemeine Eigenschaft der rationalen Entwickelungscoeficienten einer bestimmten Gattung analytischer Funktionen, J. Reine Angew. Math. 41 (1851), 368-372.
- S. Lang, Cyclotomic Fields I and II, Combined 2nd ed., Springer-Verlag, New York, 1990.
- G. Liu, Identities and congruences involving higher-order Euler-Bernoulli numbers and polynomials, Fibonacci Quart. 39 (2001), 279-284.
- G. Liu, Congruences for higher-order Euler numbers, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 30-33. https://doi.org/10.3792/pjaa.82.30
- H. Liu and W. Wang, Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums, Discrete Math. 309 (2009), 3346-3363. https://doi.org/10.1016/j.disc.2008.09.048
- H. Maiga, Some identities and congruences concerning Euler numbers and polynomials, J. Number Theory 130 (2010), 1590-1601. https://doi.org/10.1016/j.jnt.2010.01.019
- H. Maiga, New identities and congruences for Euler numbers, Advances in nonArchimedean analysis, 139-158, Contemp. Math., 665, Amer. Math. Soc., Providence, RI, 2016.
- N.E. Norlund, Vorlesungen uber Differenzenrechnung, Springer-Verlag, Berlin, 1924; Reprinted byChelsea, Bronx, New York, 1954.
- Ju.V. Osipov, p-adic zeta functions, Uspekhi Mat. Nauk 34 (1979), 209-210.
- J.L. Raabe, Zuruckfuhrung einiger Summen und bestmmtiem Integrale auf die JacobBernoullische Function, J. Reine Angew. Math. 42 (1851), 348-367.
- 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
- C.S. Ryoo, Some properties of degenerate Carlitz-type twisted q-Euler numbers and polynomials, J. Appl. Math. Inform. 39 (2021), 1-11. https://doi.org/10.14317/JAMI.2021.001
- K. Shiratani and S. Yamamoto, On a p-adic interpolation function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Ser. A 39 (1985), 113-125.
- Y. Simsek, Complete sum of products of (h, q)-extension of Euler polynomials and numbers, J. Difference Equ. Appl. 16 (2010), 1331-1348. https://doi.org/10.1080/10236190902813967
- 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
- B.A. Tangedal and P.T. Young, On p-adic multiple zeta and log gamma functions, J. Number Theory 131 (2011), 1240-1257. https://doi.org/10.1016/j.jnt.2011.01.010