mathematics Article Explicit Construction of the Inverse of an Analytic Real Function: Some Applications Joaquín Moreno 1, Miguel A. López 2,* and Raquel Martínez 2 1 Department of Applied Mathematics, Superior Technical School of Building Engineering, Polytechnic University of Valencia, 46022 Valencia, Spain; jmfl[email protected] 2 SIDIS Research Group, Department of Mathematics and Institute of Applied Mathematics in Science and Engineering (IMACI), Polytechnic School of Cuenca, University of Castilla-La Mancha, 16071 Cuenca, Spain; [email protected] * Correspondence: [email protected] Received: 12 October 2020; Accepted: 27 November 2020; Published: 3 December 2020 Abstract: In this paper, we introduce a general procedure to construct the Taylor series development of the inverse of an analytical function; in other words, given y = f (x), we provide the power series that defines its inverse x = h f (y). We apply the obtained results to solve nonlinear equations in an analytic way, and generalize Catalan and Fuss–Catalan numbers. Keywords: inverse functions; Taylor series; Taylor Remainder; nonlinear equations; Catalan numbers; Fuss–Catalan numbers MSC: 40E99; 26A99; 30B10; 05C90 1. Introduction In this paper, we have taken as a basis the previous works [1,2], in which the inverse of a polynomial function is constructed, with the aim of generalizing the methods developed there to any analytic function. That is to say: given an analytic function around the point x0: ¥ (p) f (x0) p f (x) = f (x0) + ∑ (x − x0) p=1 p! where f (p)(x) is the p-th derivative of f (x), throughout these lines, we construction the function, x = h f (y), with y = f (x) and, therefore, x0 = h f (y0) and y0 = f (x0), such that: (p) ¥ h (y ) f 0 p h f (y) = h f (y0) + ∑ (y − y0) (1) p=1 p! To accomplish this task we have organized this article as follows: • In Section2, background theory is presented. • In Section3, the successive derivatives of h f are computed in an explicit way. (p) • In Section4, a bound for jh f (y0)j (see (1)) is found. • In Section5, we study the convergence and establish the radius of convergence and Taylor Remainder of series (1). • In the next two sections, we introduce some applications for solving nonlinear equations in an analytic way, and to generalize the Catalan and Fuss–Catalan numbers. Mathematics 2020, 8, 2154; doi:10.3390/math8122154 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2154 2 of 23 • In the last section, we present our conclusions. We recall that Catalan numbers sequence is defined as: 1 2n (2n)! C = 1; C = = ; n = 1, 2, ··· (2) 0 n n + 1 n n! (n + 1)! and that they satisfy the recursive formula: C = C C n ≥ n ∑ i1 i2 ; 1 . (3) i1+i2=n−1 Catalan numbers appeared for the first time in the book Quick Methods for Accurate Values of Circle Segments, by Ming Antu (1692–1763), a Chinese mathematician. In this book, he provides some trigonometric equalities and power series, in which Catalan numbers are involved. Nicolas Fuss (1755–1826) introduced, in his paper of 1791 (see [3]), the Fuss–Catalan numbers, as: 1 m n (m n)! Cm = 1; Cm = = (4) 0 n (m − 1)n + 1 n ((m − 1)n + 1)! n! fixed m ≥ 2; with n = 1, 2, ··· . Notice that, for m = 2, they coincide with Catalan numbers. Furthermore, he provided a generalization of (3): Cm = Cm ··· Cm ; fixed m ≥ 2; 8n ≥ 1 . (5) n ∑ i1 im i1+···+im=n−1 From that time forward, throughout mathematical history, Catalan numbers have made important contributions. We list some of them that we have chosen in an arbitrary manner, by way of illustration: p • Development in power series of the function f (x) = 1 − 4x (Euler (1707–1783)). • The ballot problem (Statistics and Probability), introduced for the first time in 1887 by Joseph Bertrand (1822–1900), see [4]. • In [5], the reader can find more than 200 practical combinatorial interpretations of Catalan numbers. • Binary trees (Graph Theory), see [6]. • Lattice path theory (Graph Theory), see [7]. Throughout this paper, all the necessary computational tasks have been performed with the program Wolfram Mathematica 11.2.0.0. 2. Some Recent Results In this section, we review some outcomes, previously published by the authors, (see [1,2]), which will be used in the following sections. Such a summary has been written in detail for the sake of clarity and the self-developed reading of these lines. In fact, for our goal, we will only need formulas (19)–(21). We posed the functional equation: p 2 Q(x2 ··· , xp, h(x2, ··· , xp)) = xph (x) + ··· + x2h (x) − h(x) + 1 = 0 (6) p−1 p−1 where p > 1, x = (x2, x3, ··· , xp) 2 R and h : R ! R is the unknown to solve. We proved that, if h(x) is a solution of (6), then it satisfies the first order partial derivative equation: 2 1 + (2x2 − 1)h(x) + (3x3 + 4x2 − x2)hx2 (x) + (4x4 + 6x2x3 − 2x3)hx3 (x) + ··· (7) + (pxp + 2(p − 1)x2xp−1 − (p − 2)xp−1)hxp−1 (x) + (2px2xp − (p − 1)xp)hxp (x) = 0 ¶Q for all x, such that (x, h(x)) 6= 0. ¶h Mathematics 2020, 8, 2154 3 of 23 From (7), we showed that, if h(x) is a solution of Equation (6), then the equality: (2q2 + 3q3 + ··· + pqp)! h(q2,··· ,qp)(0, ··· , 0) = (8) (q2 + 2q3 + ··· + (p − 1)qp + 1)! (q ,··· ,qp) holds, with q2 + ··· + qp = n and q2, ..., qp non-negative integers, where h 2 (0, ··· , 0) is the n-th partial derivative of h, q2 times with respect to x2, ...., qp times with respect to xp. Therefore, we can express the function h(x) as: ¥ ( ) = q2 ··· qp h x ∑ ∑ Cq2···qp x2 xp (9) n=0 q2+···+qp=n where (2 q2 + 3 q3 + ··· + p qp)! Cq2...qp = (10) (q2 + 2 q3 + ··· + (p − 1)qp + 1)! q2! q3! ··· qp! Consider the polynomial function of degree p, given by: p y = P(x) = a0 + a1x + ··· + apx (a1, ap 6= 0) (11) with x, ai 2 R, 0 ≤ i ≤ p, and the functions Xi, 2 ≤ i ≤ p: (a − y)i−1a ( ) = 0 i Xi y i (12) (−a1) then, the series: a0 − y fP(y) = h(X2(y), ··· , Xp(y)) (13) −a1 is the inverse function of P(x), if it makes sense. Taking into account (9), by substituting functions (12) in (13), we obtain: qp ¥ q2 p−1 ! a0 − y (a0 − y)a2 (a0 − y) ap f (y) = Cq q ... P −a ∑ ∑ 2... p (−a )2 (−a )p 1 n=0 q2+...+qp=n 1 1 (14) − ¥ q2 a qp y a0 q +...+(p−1) qp a2 p q +...+(p−1) qp = (−1) 2 Cq q ... (y − a ) 2 . a ∑ ∑ 2... p (−a )2 (−a )p 0 1 n=0 q2+...+qp=n 1 1 Next, by making the subscripts and superscripts change: n1 = q2 + 2q3 + ... + (p − 1) qp, the terms of series (14) are rearranged in the form: 2 3 ¥ q2 qp y − a a ap ( ) = 0 (− )n1 2 ( − )n1 fP y ∑ 4 ∑ 1 Cq2...qp 2 ... p 5 y a0 . (15) a1 (−a1) (−a1) n1=0 q2+...+(p−1) qp=n1 Series (14) and (15) are absolutely convergent in a neighborhood of a0, Va0 , given by: ( 2 p p−1 ) p (a0 − y)a2 p (a0 − y) ap Va = y 2 R; + ··· + < 1 (16) 0 p − 1 2 ( − )p−1 p a1 p 1 a1 that is, obviously, not empty. From now on, we will denote the inverse function of y = P(x) as x = hP(y). Therefore, if the inequality: 2 3 2 p p−1 p a a p a a p a ap 0 2 + 0 3 + ··· + 0 < 1 (17) p − 1 2 (p − 1)2 3 ( − )p−1 p a1 a1 p 1 a1 Mathematics 2020, 8, 2154 4 of 23 holds then, the series: − qp ¥ q2 p 1 ! a a a a ap r = 0 C 0 2 ... 0 (18) −a ∑ ∑ q2...qp (−a )2 (−a )p 1 n=0 q2+...+qp=n 1 1 is absolutely convergent to the root of P(x), r, closest to the coordinate origin. 0 Finally, we consider the Taylor series of the polynomial, P(x), around the point x0, with P (x0) 6= 0, which is: P00(x ) P(p)(x ) y = P(x) = P(x ) + P0(x )(x − x ) + 0 (x − x )2 + ... + 0 (x − x )p (19) 0 0 0 2! 0 p! 0 with y0 = P(x0). Then, one gets: 2 y − y ¥ P00(x ) q2 ( ) = + 0 (− )n1 0 hP y x0 0 ∑ 4 ∑ 1 Cq2...qp 0 2 ... P (x0) 2! (−P (x0)) n1=0 q2+2 q3+...+(p−1) qp=n1 q (p) ! p # P (x0) n1 ··· 0 p y − y0 p! (−P (x0)) 2 (20) 1 ¥ P00(x ) q2 = + (− )n1 0 x0 0 ∑ 4 ∑ 1 Cq2...qp 0 2 ... P (x0) 2! (−P (x0)) n1=0 q2+2 q3+...+(p−1) qp=n1 !qp # P(p)(x ) 0 n1+1 ··· 0 p (y − y0) p! (−P (x0)) hP(y) being the inverse function of (19) in Vy0 : ( 2 00 p p−1 (p) ) p (y0 − y)P (x0) p (y0 − y) P (x0) V = y 2 + ··· + < y0 R; 0 2 p−1 0 p 1 (21) p − 1 2! P (x0) (p − 1) p! P (x0) according to (16), with y0 = P(x0). Once we have seen the background on which this paper is based, we introduce the original contributions of this article in the coming sections. 3. Definition of the Function h f and Calculation of Its Derivatives Theorem 1.