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MONOTONIC PHINOMIAL COEFFICIENTS

Part of: Elementary number theory Sequences and sets

Published online by Cambridge University Press:  09 January 2017

FLORIAN LUCA  and
PANTELIMON STĂNICĂ
FLORIAN LUCA*
Affiliation:
School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits 2050, South Africa email florian.luca@wits.ac.za
PANTELIMON STĂNICĂ
Affiliation:
Naval Postgraduate School, Applied Mathematics Department, Monterey, CA 93943-5216, USA Institute of Mathematics ‘Simion Stoilow’, Romanian Academy, Bucharest, Romania email pstanica@nps.edu
*
✉florian.luca@wits.ac.za
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Abstract

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We investigate the monotonic characteristics of the generalised binomial coefficients (phinomials) based upon Euler’s totient function. We show, unconditionally, that the set of integers for which this sequence is unimodal is finite and, assuming the generalised Riemann hypothesis, we find all the exceptions.

Keywords

Euler functiongeneralised binomial coefficientsmonotony

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11B25: Arithmetic progressions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 365 - 372
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

References

Edgar, T., ‘Totienomial coefficients’, Integers 14 (2014), Art. A62.Google Scholar
Edgar, T. and Spivey, M. Z., ‘Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers’, J. Integer Seq. 19 (2016), Art. 16.1.6.Google Scholar
Knuth, D. E. and Wilf, H. S., ‘The power of a prime that divides a generalized binomial coefficient’, J. reine angew. Math. 396 (1989), 212–219.Google Scholar
Rosser, J. B. and Schoenfeld, L., ‘Approximate formulas for some functions of prime numbers’, Illinois J. Math. 6 (1962), 64–94.CrossRefGoogle Scholar
Washington, L., Introduction to Cyclotomic Fields, 2nd edn (Springer, New York, 1997).CrossRefGoogle Scholar
Winckler, B., ‘Théorème de Chebotarev effectif’, Preprint, 2013, arXiv:1311.5715.Google Scholar