Euler–Worpitzky–Chen polynomials

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In mathematics, the Euler–Worpitzky–Chen polynomials are defined as

W_n(x) = \sum_{k=0}^{n} \sum_{v=0}^{k} (-1)^{v} \binom{k}{v} c_k (x+v)^n \ ,

where the Chen sequence ck is defined for k ≥ 0 as

c_k = \frac{(-1)^{\left\lfloor k/4 \right\rfloor} } {2^{\left\lfloor k/2 \right\rfloor}} [4 \nmid k] = 1,1,\frac12,0,-\frac14,-\frac14,-\frac18,0,\ldots \ .

The expression [4 notdiv k] has the value 0 if 4 divides k and 1 otherwise.

The Euler-Worpitzky-Chen polynomials were introduced in 2008. The first few are

W0(x) =  1        
W1(x) =  x − 1       
W2(x) =  x2  − x 2      
W3(x) =  x3  − x2 3 + 2    
W4(x) =  x4  − x3 4 + x 8    
W5(x) =  x5  − x4 5 + x2 20 − 16  
W6(x) =  x6  − x5 6 + x3 40 − x 96  
W7(x) =  x7  − x6 7 + x4 70 − x2 336 + 272

The coefficients of the Euler-Worpitzky-Chen polynomials are integers, in contrast to the coefficients of the Euler and Bernoulli polynomials, which are rational numbers.

  • Wn(1) = En the Euler numbers.
  • Wn(0) = Tn are the tangent numbers.
  • Wn-1(0) n / (2n − 4n) = Bn gives for n > 1 the Bernoulli numbers.
  • 2n Wn(1/2) = 1,-1,-3,11,57,-361,-2763,... , are the generalized Euler numbers, or Springer numbers, sequence A001586 in the Encyclopedia of Integer Sequences.

[edit] The sinusoidal character of the polynomials.

The scaled Euler-Worpitzky-Chen polynomials are defined as

\omega_n(x) = W_n(x+1)\,/\,max(|W_n(0)|,|W_n(1)|)\ .

Plotting ωn(x) shows the sinusoidal behavior of these polynomials, which is easily overlooked in the nonscaled form. For odd index ωn(x) approximates ±sin(xπ/2) and for even index ±cos(xπ/2) in an interval enclosing the origin. This observation expands the observation that the Euler and Bernoulli number have π as a common root to an continuous scale.

But much more is true: the domain of sinusoidal behavior gets larger and larger as the degree of the polynomials grows. In fact ωn(x) shows, in an asymptotical precise sense, sinusoidal behavior in the interval [-2n/πe, 2n/πe].

From these observations follows the regular behavior of the real roots of the Euler-Worpitzky-Chen polynomials. For example the roots of ωn(x) are close to the integer lattices: {±0,±2,±4,...} if n is odd and {±1,±3,±5,...} if n is even.

[edit] References

  • J. Worpitzky, "`Studien über die Bernoullischen und Eulerschen Zahlen."', Journal für die reine und angewandte Mathematik, 94 (1883), 203--232.
  • Kwang-Wu Chen, "`Algorithms for Bernoulli numbers and Euler numbers."', Journal of Integer Sequences, 4 (2001), [01.1.6].