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Applications of an explicit formula for the generalized Euler numbers. (English) Zbl 1145.11020
For real number \(x\) and positive integers \(n,k\), let \(E^{(x)}_{2n}\), \(s(n,k)\), \(T(n,k)\) denote the generalized Euler numbers, Stirling numbers and central factorial numbers, respectively. In the present paper under review, the authors prove an explicit formula for \(E^{(x)}_{2n}\) as follows: \[ E^{(x)}_{2n}=\sum_{i=1}^n\rho(n,i)x^i, \] here
\[ \rho(n,k)=(-1)^k\sum_{j=k}^n\frac{(2j)!}{2^jj!}s(j,k)T(n,j). \] By using this formula, they also obtain some interesting identities and congruences involving the higher-order Euler numbers, Stirling numbers, the central factorial numbers and values of the Riemann zeta function.

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11B73 Bell and Stirling numbers
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11B83 Special sequences and polynomials
05A10 Factorials, binomial coefficients, combinatorial functions
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[1] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F. G.: Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, 1953 · Zbl 0052.29502
[2] Luke, Y. L.: The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969 · Zbl 0193.01701
[3] Srivastava, H. M., Choi, J.: Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001 · Zbl 1014.33001
[4] Srivastava, H. M.: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Cambridge Philos. Soc., 129, 77–84 (2000) · Zbl 0978.11004 · doi:10.1017/S0305004100004412
[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) · Zbl 0957.11036 · doi:10.1006/jmaa.2000.6746
[6] Srivastava, H. M., Pintér, Á.: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett., 17, 375–380 (2004) · Zbl 1070.33012 · doi:10.1016/S0893-9659(04)90077-8
[7] Zhang, W. P.: Some identities involving the Euler and the central factorial numbers. Fibonacci Quart., 36, 154–157 (1998) · Zbl 0919.11018
[8] Liu, G. D.: Summation and recurrence formula involving the central factorial numbers and zeta function. Appl. Math. Comput., 149, 175–186 (2004) · Zbl 1100.11027 · doi:10.1016/S0096-3003(02)00964-5
[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) · Zbl 1039.11008
[10] Liu, G. D.: The solution of problem for Euler numbers. Acta Mathematica Sinica, Chinese Series, 47, 825–828 (2004) · Zbl 1130.11308
[11] Liu, G. D.: On congruences of Euler numbers modulo and odd square. Fibonacci Quart., 43, 132–136 (2005) · Zbl 1165.11309
[12] Liu, G. D.: Congruences for higher-order Euler numbers. Proc. Japan Acad. Ser.A, 82, 30–33 (2006) · Zbl 1120.11011 · doi:10.3792/pjaa.82.30
[13] Nörlund, N. E.: Vorlesungenüber Differenzenrechnung, Springer-Verlag, Berlin 1924; Reprinted by Chelsea Publishing Company, Bronx, New York, 1954 · JFM 50.0315.02
[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 · Zbl 0283.05001
[15] Jordan, C.: Calculus of Finite Differences, New York, Chelsea, 1965 · Zbl 0154.33901
[16] Riordan, J.: Combinatorial Identities, New York, Wiley, 1968 · Zbl 0194.00502
[17] Apostol, T. M.: Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 · Zbl 0335.10001
[18] Berndt, B. C.: Classical theorems on quadratic residues. Enseign. Math. II, 22, 261–304 (1976) · Zbl 0337.10031
[19] Kummer, E. E.: Über eine allgemeine Eigenschaft der rationalen Entwickelungscoëfficienten einer bestimmten Gattung analytischer Functionen. J. Reine Angew. Math., 41, 368–372 (1851) · ERAM 041.1136cj · doi:10.1515/crll.1851.41.368
[20] Guy, R. K.: Unsolved problem in number theory, 2nd. ed., Springer-Verlag, New York, 1994 · Zbl 0805.11001
[21] Yuan, P. Z.: A conjecture on Euler numbers. Proc. Japan Acad. Ser.A, 80, 180–181 (2004) · Zbl 1080.11018 · doi:10.3792/pjaa.80.180
[22] Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990 · Zbl 0712.11001
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