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.
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
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 |
Citations:
Zbl 0689.00009Digital Library of Mathematical Functions:
Szegő–Szász Inequality ‣ §18.14(iii) Local Maxima and Minima ‣ §18.14 Inequalities ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal PolynomialsIn §18.4(i) Graphs ‣ §18.4 Graphics ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
In §18.4 Graphics ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials
