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On the higher derivatives of the inverse tangent function

In this paper, we find explicit formulas for higher-order derivatives of the inverse tangent function. More precisely, we study polynomials that are induced from the higher-order derivatives of arctan(x). Successively, we give generating functions, recurrence relations, and some particular properties for these polynomials. | Turk J Math (2018) 42: 2643 – 2656 © TÜBİTAK doi:10.3906/mat-1712-40 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article On the higher derivatives of the inverse tangent function Mohamed Amine BOUTICHE∗,, Mourad RAHMANI, Faculty of Mathematics, USTHB, Algiers, Algeria Received: 13.12.2017 • Accepted/Published Online: 31.07.2018 • Final Version: 27.09.2018 Abstract: In this paper, we find explicit formulas for higher-order derivatives of the inverse tangent function. More precisely, we study polynomials that are induced from the higher-order derivatives of arctan(x) . Successively, we give generating functions, recurrence relations, and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas, and matching polynomials are established. Key words: Explicit formula, derivative polynomial, inverse tangent function, Chebyshev polynomial, matching polynomial 1. Introduction The problem of establishing closed formulas for the n -derivative of the function arctan(x) is not straightforward and has been proved to be important for deriving rapidly convergent series for π [2, 3, 14]. Recently, many authors investigated the aforementioned problem and derived simple explicit closed-form higher derivative formulas for some classes of functions. In [1, 6, 8] and references therein, the authors found explicit forms of the derivative polynomials of the hyperbolic, trigonometric tangent, cotangent, and secant functions. Several new closed formulas for higher-order derivatives have been established for trigonometric and hyperbolic functions in [19], tangent and cotangent functions in [16], and arc-sine functions in [17]. We note from entries 1.1.7(3) and 1.1.7(4) in chapter 1 of Brychkov’s handbook [7, p. 14] that the higher-order derivatives of arctan(x) can be expressed in terms of Chebyshev polynomials as follows: ( ) ( ) d2n n 2n+1 2 2 −n−1/2 √ 1 (arctan(ax)) = (−1) (2n − 1)!a x 1 + a x U (n ≥ .