Askey, Richard A. Graphs as an aid to understanding special functions. (English) Zbl 0694.33002 Asymptotic and computational analysis. Conference in honor of Frank W.J. Olver’s 65th birthday, Proc. Int. Symp., Winnipeg/Can. 1989, Lect. Notes Pure Appl. Math. 124, 3-33 (1990). [For the entire collection see Zbl 0689.00009.] Graphs can play an important role in suggesting inequalities for special functions. Some classical examples are given, including Todd’s observation about the monotonicity of relative maxima of adjacent Legendre polynomials. A new proof is given of this theorem of Szegö. A similar inequality holds for Legendre functions of the second kind \(Q_ n(x)\). This is suggested by a graph in Jahnke and Emde, and proven in a later paper. Reviewer: R.A.Askey Cited in 4 Documents MSC: 33C05 Classical hypergeometric functions, \({}_2F_1\) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 26D05 Inequalities for trigonometric functions and polynomials Keywords:Legendre polynomials; inequalities Citations:Zbl 0689.00009 PDF BibTeX XML OpenURL Digital Library of Mathematical Functions: Szegő–Szász Inequality ‣ §18.14(iii) Local Maxima and Minima ‣ §18.14 Inequalities ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials Figure 18.4.2 ‣ §18.4(i) Graphs ‣ §18.4 Graphics ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials Figure 18.4.2 ‣ §18.4 Graphics ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials