Bernoulli Polynomials - Maximum and Minimum

Maximum and Minimum

At higher n, the amount of variation in Bn(x) between x = 0 and x = 1 gets large. For instance,

B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\frac{182}{3}x^{12}+\frac{572}{3}x^{10}-429x^8+\frac{1820}{3}x^6 -\frac{1382}{3}x^4+140x^2-\frac{3617}{510}

which shows that the value at x = 0 (and at x = 1) is −3617/510 ≈ −7.09, while at x = 1/2, the value is 118518239/3342336 ≈ +7.09. D.H. Lehmer showed that the maximum value of Bn(x) between 0 and 1 obeys

unless n is 2 modulo 4, in which case

(where is the Riemann zeta function), while the minimum obeys

unless n is 0 modulo 4, in which case

These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well.

Read more about this topic:  Bernoulli Polynomials

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