Graphs as an aid to understanding special functions.

*(English)*Zbl 0694.33002Asymptotic 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 ${Q}_{n}\left(x\right)$. This is suggested by a graph in Jahnke and Emde, and proven in a later paper.

Reviewer: R.A.Askey