# On Kaluza's sign criterion for reciprocal power series

Árpád Baricz; Jetro Vesti; Matti Vuorinen

Annales UMCS, Mathematica (2011)

- Volume: 65, Issue: 2, page 1-16
- ISSN: 2083-7402

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topÁrpád Baricz, Jetro Vesti, and Matti Vuorinen. "On Kaluza's sign criterion for reciprocal power series." Annales UMCS, Mathematica 65.2 (2011): 1-16. <http://eudml.org/doc/267915>.

@article{ÁrpádBaricz2011,

abstract = {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.},

author = {Árpád Baricz, Jetro Vesti, Matti Vuorinen},

journal = {Annales UMCS, Mathematica},

keywords = {Power series; log-convexity; hypergeometric functions; trigonometric functions},

language = {eng},

number = {2},

pages = {1-16},

title = {On Kaluza's sign criterion for reciprocal power series},

url = {http://eudml.org/doc/267915},

volume = {65},

year = {2011},

}

TY - JOUR

AU - Árpád Baricz

AU - Jetro Vesti

AU - Matti Vuorinen

TI - On Kaluza's sign criterion for reciprocal power series

JO - Annales UMCS, Mathematica

PY - 2011

VL - 65

IS - 2

SP - 1

EP - 16

AB - 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.

LA - eng

KW - Power series; log-convexity; hypergeometric functions; trigonometric functions

UR - http://eudml.org/doc/267915

ER -

## References

top- Anderson, G. D., Vamanamurthy, M. K. and Vuorinen, M., Conformal Invariants, Inequalities and Quasiconformal Maps, John Wiley & Sons, Inc., New York, 1997. Zbl0885.30012
- Baricz, Á., Generalized Bessel Functions of the First Kind, Lecture Notes in Mathematics, vol. 1994, Springer-Verlag, Berlin, 2010. Zbl1203.33001
- Baricz, Á., Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. 53 (3) (2010), 575-599.[WoS][Crossref] Zbl1202.33008
- Beckenbach, E. F., Bellman, R., Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 30, Springer-Verlag, Berlin, 1961.
- Belzunce, F., Ortega, E. M. and Ruiz, J. M., On non-monotonic ageing properties from the Laplace transform, with actuarial applications, Insurance Math. Econom. 40 (2007), 1-14.[Crossref] Zbl1273.91231
- Biernacki, M., Krzyż, J., On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A. 9 (1955), 135-147. Zbl0078.26402
- Carlitz, L., Advanced Problems and Solutions: Solutions: 4803, Amer. Math. Monthly 66 (5) (1959), 239-240.
- Davenport, H., Pólya, G., On the product of two power series, Canadian J. Math. 1 (1949), 1-5. Zbl0037.32505
- Hansen, B. G., Steutel, F. W., On moment sequences and infinitely divisible sequences, J. Math. Anal. Appl. 136 (1988), 304-313.[Crossref] Zbl0665.60022
- Hardy, G. H., Divergent Series, With a preface by J. E. Littlewood and a note by L. S. Bosanquet. Reprint of the revised (1963) edition. Éditions Jacques Gabay, Sceaux, 1992. Zbl0897.01044
- Heikkala, V., Vamanamurthy, M. K. and Vuorinen, M., Generalized elliptic integrals, Comput. Methods Funct. Theory 9 (1) (2009), 75-109.[Crossref]
- Henrici, P., Applied and computational complex analysis. Vol. 1, Power series - integration - conformal mapping - location of zeros, Reprint of the 1974 original, John Wiley & Sons, Inc., New York, 1988. Zbl0635.30001
- Horn, R. A., On moment sequences and renewal sequences, J. Math. Anal. Appl. 31 (1970), 130-135.[Crossref] Zbl0175.16701
- Jurkat, W. B., Questions of signs in power series, Proc. Amer. Math. Soc. 5 (6) (1954), 964-970.[Crossref] Zbl0059.06105
- Kaluza, T., Über die Koeffizienten reziproker Potenzreihen, Math. Z. 28 (1928), 161-170.[Crossref] Zbl54.0335.03
- Kendall, D. G., Renewal sequences and their arithmetic, Proceedings of Loutraki Symposium on Probability Methods in Analysis, Lecture Notes in Mathematics, vol. 31, Springer-Verlag, New York-Berlin, 1967, pp. 147-175. Zbl0178.19803
- Lamperti, J., On the coefficients of reciprocal power series, Amer. Math. Monthly 65 (2) (1958), 90-94.[Crossref] Zbl0081.29503
- Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (eds.), NIST Handbook of Mathematical Functions, Cambridge Univ. Press, Cambridge, 2010. Zbl1198.00002
- Szegő, G., Bemerkungen zu einer Arbeit von Herrn Fejér über die Legendreschen Polynome, Math. Z. 25 (1926), 172-187.[Crossref] Zbl52.0351.03

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