Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. (English) Zbl 1076.33006

The authors introduce a generalization of the classical Bernoulli polynomials as analogous definition of Apostol type [see T. M. Apostol, Pac. J. Math. 1, 161–167 (1951; Zbl 0043.07103)] for the so-called Apostol-Bernoulli numbers and polynomials of higher order. The generalization, the Apostol-Bernoulli polynomials \({\mathcal B}_n^{(\alpha)}(x;\lambda)\), is defined by means of the following generating function: \[ \left(\frac{z}{\lambda\,e^z-1}\right)^{\alpha}\,e^{xz} =\sum_{n=0}^\infty {\mathcal B}_n^{(\alpha)}(x;\lambda)\,\frac{z^n}{n!}\qquad \left(| z+\log \lambda| <2\pi;\,1^{\alpha}:=1\right) \] with \[ B_{n}^{(\alpha)}(x)={\mathcal B}_n^{(\alpha)}(x;1) \quad\text{and}\quad {\mathcal B}_{n}^{(\alpha)}(\lambda):={\mathcal B}_n^{(\alpha)}(0;\lambda) \] where \({\mathcal B}_{n}^{(\alpha)}(\lambda)\) denotes the so-called Apostol-Bernoulli numbers of order \(\alpha\). In a similar manner the Apostol-Euler polynomials of order \(\alpha\), a generalization of the classical Euler polynomials, is introduced. In a previous paper, the first author derived several properties and explicit representations of the Apostol-Euler polynomials of order \(\alpha\). In this paper, the authors investigate the corresponding problems for the Apostol-Bernoulli polynomials of order \(\alpha\) by following the work of the second author in an earlier article [see H. M. Srivastava, Math. Proc. Camb. Philos. Soc. 129, 77–84 (2000; Zbl 0978.11004)]. They establish their elementary properties, derive an explicit series representations for the polynomials involving the Gaussian hypergeometric function, the Hurwitz zeta function and the Riemann zeta function.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B68 Bernoulli and Euler numbers and polynomials
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[2] Apostol, T.M., On the lerch zeta function, Pacific J. math., 1, 161-167, (1951) · Zbl 0043.07103
[3] Apostol, T.M., Introduction to analytic number theory, (1976), Springer-Verlag New York · Zbl 0335.10001
[4] Comtet, L., Advanced combinatorics: the art of finite and infinite expansions, (1974), Reidel Dordrecht, Translated from the French by J.W. Nienhuys
[5] Cvijović, D.; Klinowski, J., New formulae for the Bernoulli and Euler polynomials at rational arguments, Proc. amer. math. soc., 123, 1527-1535, (1995) · Zbl 0827.11012
[6] Luo, Q.-M., On the apostol – bernoulli polynomials, Central European J. math., 2, 509-515, (2004) · Zbl 1073.33001
[7] Q.-M. Luo, Apostol-Euler polynomials of higher order and the Gaussian hypergeometric function, Taiwanese J. Math., in press
[8] Nörlund, N.E., Vorlesungen über differentzenrechnung, (1924), Springer-Verlag Berlin, Reprinted by Chelsea, Bronx, New York, 1954 · JFM 50.0315.02
[9] 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
[10] Srivastava, H.M.; Choi, J., Series associated with the zeta and related functions, (2001), Kluwer Academic Dordrecht · Zbl 1014.33001
[11] Srivastava, H.M.; Todorov, P.G., An explicit formula for the generalized Bernoulli polynomials, J. math. anal. appl., 130, 509-513, (1988) · Zbl 0621.33008
[12] Todorov, P.G., Une formule simple explicite des nombres de Bernoulli généralisés, C. R. acad. sci. Paris Sér. I math., 301, 665-666, (1985) · Zbl 0606.10008
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