Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine

• Published in: Open mathematics (Warsaw, Poland) Vol. 22; no. 1; pp. 3130 - 3144
• Main Authors: Pei, Wei-Juan, Guo, Bai-Ni
• Format: Journal Article
• Language: English
• Published: Warsaw De Gruyter 02.12.2024; De Gruyter Poland
• Subjects: 11B68; 15A15; 26A06; 26A09; 26A51; 33B10; 41A58; Bernoulli number; Convexity; cosine; derivative formula; Hypergeometric functions; logarithmic convexity; Logarithms; MacLaurin series; Maclaurin series expansion; monotonicity; monotonicity rule; Qi’s normalized remainder; Quotients; ratio; Series expansion; Trigonometric functions
• ISSN: 2391-5455; 2391-5455
• DOI: 10.1515/math-2024-0095

In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.