Abstract
We present a rational extension of the Newton diagram for the positivity of \({}_1F_2\) generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots \(\beta (\alpha )\) of
where \(j_{\alpha , 2}\) denotes the second positive zero of Bessel function \(J_\alpha \).
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Acknowledgements
We are deeply grateful to Professor George Gasper for providing us a much simpler proof for the improved part of the Askey–Szegö problem and an insightful note on Whipple’s transformation formula which are presented in Sect. 6. Yong-Kum Cho is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1A09083148).
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Communicated by Mourad Ismail.
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Cho, YK., Chung, SY. & Yun, H. Rational Extension of the Newton Diagram for the Positivity of \({}_1F_2\) Hypergeometric Functions and Askey–Szegö Problem. Constr Approx 51, 49–72 (2020). https://doi.org/10.1007/s00365-019-09462-5
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DOI: https://doi.org/10.1007/s00365-019-09462-5