Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4582614 | Expositiones Mathematicae | 2007 | 6 Pages |
Abstract
LetÎn(x)=Pn(x)2-Pn-1(x)Pn+1(x),where Pn is the Legendre polynomial of degree n. A classical result of Turán states that În(x)⩾0 for xâ[-1,1] and n=1,2,3,â¦. Recently, Constantinescu improved this result. He establishedhnn(n+1)(1-x2)⩽În(x)(-1⩽x⩽1;n=1,2,3,â¦),where hn denotes the nth harmonic number. We present the following refinement. Let n⩾1 be an integer. Then we have for all xâ[-1,1]αn(1-x2)⩽În(x)with the best possible factorαn=μ[n/2]μ[(n+1)/2].Here, μn=2-2n2nn is the normalized binomial mid-coefficient.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Horst Alzer, Stefan Gerhold, Manuel Kauers, Alexandru LupaÅ,