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A Look at Generalized Trigonometric Functions as Functions of Their Two Parameters and Further New Properties

Author

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  • Dmitrii Karp

    (Department of Mathematics, Holon Institute of Technology, 52 Golomb Street, P.O. Box 305, Holon 5810201, Israel
    School of Economics and Management, Far Eastern Federal University, 690922 Vladivostok, Russia
    These authors contributed equally to this work.)

  • Elena Prilepkina

    (Institute of Applied Mathematics, Far East Branch of the Russian Academy of Sciences (FEBRAS), 690041 Vladivostok, Russia
    These authors contributed equally to this work.)

Abstract

Investigation of the generalized trigonometric and hyperbolic functions containing two parameters has been a very active research area over the last decade. We believe, however, that their monotonicity and convexity properties with respect to parameters have not been thoroughly studied. In this paper, we make an attempt to fill this gap. Our results are not complete; for some functions, we manage to establish (log)-convexity/concavity in parameters, while for others, we only managed the prove monotonicity, in which case we present necessary and sufficient conditions for convexity/concavity. In the course of the investigation, we found two hypergeometric representations for the generalized cosine and hyperbolic cosine functions which appear to be new. In the last section of the paper, we present four explicit integral evaluations of combinations of generalized trigonometric/hyperbolic functions in terms of hypergeometric functions.

Suggested Citation

  • Dmitrii Karp & Elena Prilepkina, 2024. "A Look at Generalized Trigonometric Functions as Functions of Their Two Parameters and Further New Properties," Mathematics, MDPI, vol. 12(21), pages 1-17, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:21:p:3383-:d:1509449
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