## 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.
 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