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On Kaluza's sign criterion for reciprocal power series

Árpád Baricz, Jetro Vesti, Matti Vuorinen (2011)

Annales UMCS, Mathematica

T. Kaluza has given a criterion for the signs of the power series of a function that is the reciprocal of another power series. In this note the sharpness of this condition is explored and various examples in terms of the Gaussian hypergeometric series are given. A criterion for the monotonicity of the quotient of two power series due to M. Biernacki and J. Krzyż is applied.

On the apostol-bernoulli polynomials

Qiu-Ming Luo (2004)

Open Mathematics

In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.

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