Applications of an explicit formula for the generalized Euler Numbers

Acta Mathematica Sinica, English Series

Volumn 24, Issue 2, 2008, Pages 343-352

  Liu, Guo Dong ^a   Zhang, Wen Peng ^b  

^a HUIZHOU UNIVERSITY   (China)

^b NORTHWEST UNIVERSITY   (China)

Author keywords

Congruences; Dirichlet L function; Stirling numbers; The central factorial numbers; The generalized Euler numbers

Indexed keywords

EID: 40549114005     PISSN: 14398516     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10114-007-1013-x     Document Type: Article

References (22)

1. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, 1953
2. Luke, Y. L.: The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969
3. Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001
4. Srivastava, H. M.: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Cambridge Philos. Soc., 129, 77-84 (2000)
5. Srivastava, H. M.: Some simple algorithms for the evaluations and representations of the Riemann zeta function at positive integer arguments. J. Math. Anal, Appl., 246, 331-351 (2000)
6. Srivastava, H. M., Pintér, Á.: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett., 17, 375-380 (2004)
7. Zhang, W. P.: Some identities involving the Euler and the central factorial numbers. Fibonacci Quart., 36, 154-157 (1998)
8. Liu, G. D.: Summation and recurrence formula involving the central factorial numbers and zeta function. Appl. Math. Comput., 149, 175-186 (2004)
9. Liu, G. D.: The generalized central factorial numbers and higher order Nörlund Euler-Bernoulli polynomials. Acta Mathematica Sinica, Chinese Series, 44, 933-946 (2001)
10. Liu, G. D.: The solution of problem for Euler numbers. Acta Mathematica Sinica, Chinese Series, 47, 825-828 (2004)
11. Liu, G. D.: On congruences of Euler numbers modulo and odd square. Fibonacci Quart., 43, 132-136 (2005)
12. Liu, G. D.: Congruences for higher-order Euler numbers. Proc. Japan Acad. Ser.A, 82, 30-33 (2006)
13. Nörlund, N. E.: Vorlesungenüber Differenzenrechnung, Springer-Verlag, Berlin 1924; Reprinted by Chelsea Publishing Company, Bronx, New York, 1954
14. Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys), Reidel, Dordrecht and Boston, 1974
15. Jordan, C.: Calculus of Finite Differences, New York, Chelsea, 1965
16. Riordan, J.: Combinatorial Identities, New York, Wiley, 1968
17. Apostol, T. M.: Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976
18. Berndt, B. C.: Classical theorems on quadratic residues. Enseign. Math. II, 22, 261-304 (1976)
19. Kummer, E. E.: Über eine allgemeine Eigenschaft der rationalen Entwickelungscoëfficienten einer bestimmten Gattung analytischer Functionen. J. Reine Angew. Math., 41, 368-372 (1851)
20. Guy, R. K.: Unsolved problem in number theory, 2nd. ed., Springer-Verlag, New York, 1994
21. Yuan, P. Z.: A conjecture on Euler numbers. Proc. Japan Acad. Ser.A, 80, 180-181 (2004)
22. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990