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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

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