×

Bernoulli numbers and polynomials of arbitrary complex indices. (English) Zbl 0768.11010

For complex \(\alpha\) with \(\text{Re }\alpha > 1\) the authors define the Bernoulli periodic function \({\mathcal B}_ \alpha(x)\) with period 1 by the Fourier series \[ {\mathcal B}_ \alpha = -2\Gamma(\alpha + 1)\sum^ \infty_{k=1}{\cos(2\pi kx - \alpha\pi/2)\over (2\pi k)^ \alpha}, \] and study its connection with the classical Bernoulli polynomials and Bernoulli numbers.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berndt, B. C., Ramanujan’s Notebook, Part I (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0555.10001
[2] Butzer, P. L.; Hauss, M.; Schmidt, M., Factorial functions and Stirling numbers of fractional orders, Resultate Math., 16, 16-48 (1989) · Zbl 0707.05002
[3] Butzer, P. L.; Hauss, M., On Stirling functions of the second kind, Stud. Appl. Math., 84, 71-91 (1991) · Zbl 0738.11025
[4] Butzer, P. L.; Hauss, M., Riemann Zeta function: Rapidly converging series and integral representations, Appl. Math. Lett., 5, 83-88 (1992) · Zbl 0746.11034
[5] P.L. Butzer and M. Hauss, Eulerian numbers with fractional order parameters, Aequationes Math. (to appear).; P.L. Butzer and M. Hauss, Eulerian numbers with fractional order parameters, Aequationes Math. (to appear). · Zbl 0797.11025
[6] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for the Special Functions of Mathematical Physics (1966), Springer-Verlag: Springer-Verlag Berlin · Zbl 0143.08502
[7] Apostol, T. M., Introduction to Analytic Number Theory (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0335.10001
[8] Butzer, P. L.; Nessel, R. J., Fourier Analysis and Approximation (1971), Birkhäuser and Academic Press: Birkhäuser and Academic Press Basel and New York · Zbl 0217.42603
[9] Abou-Tair, I. A., On a certain class of Dirichlet series, Appl. Anal., 35, 205-219 (1990) · Zbl 0668.30004
[10] Berndt, B. C., On the Hurwitz Zeta-function, Rocky Mountain J. Math., 2, 151-157 (1972) · Zbl 0229.10023
[11] Apostol, T. M., Some series involving the Reimann zeta function, Proc. Amer. Math. Soc., 5, 239-243 (1954) · Zbl 0055.06903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.