Euler–Worpitzky–Chen polynomials

= Introduction =

The Euler-Worpitzky-Chen polynomials are closely related to the family of Euler-Bernoulli polynomials and numbers. The coefficients of the polynomialsare integers, in contrast to the coefficients of the Euler and Bernoulli polynomials, which are rational numbers. Thus the Euler-Worpitzky-Chen polynomials can be computed more easily and applied to special number theory.The Euler, Bernoulli, Genocchi, Euler zeta, tangent as well as the up-down numbers and the Springer numbers are either values or scaled values of these polynomials. The Euler-Worpitzky-Chen polynomials are also connected withthe Riemann and Hurwitz zeta function. They display a strong and beautiful sinusoidal behavior if properly scaled which has its basis in the Fourier analysis of the generalized zeta function.

The Euler-Worpitzky-Chen polynomials are possibly new. The present author learned them from Peter Luschny in 2008. The polynomials are named after Leonhard Euler, Julius Worpitzky, and Kwang-Wu Chen.

Definition

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

:

where the Chen sequence "c"_"k" is definedfor "k" 0 as

: $c\_k\; =\; frac\{(-1)^\{leftlfloor\; k/4\; ight\; floor\}\; \}\; \{2^\{leftlfloor\; k/2\; ight\; floor\; [4\; mid\; k]\; =\; 1,1,frac12,0,-frac14,-frac14,-frac18,0,ldots\; .$

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

The first few Euler-Worpitzky-Chen polynomials are displayed in the next table.

W0(x) =	1,
W1(x) =	x,
W2(x) =	x2	1,
W3(x) =	x3	3x,
W4(x) =	x4	6x2	+ 5,
W5(x) =	x5	10x3	+ 25x,
W6(x) =	x6	15x4	+ 75x2	61,
W7(x) =	x7	21x5	+ 175x3	427x.

Application

The Euler-Worpitzky-Chen polynomialscan be applied to the computation of a variety of classical numbers.

• "W"_"n"(0) = "E"_"n" are the Euler numbers 1,0,-1,0,5,0,-61,....
• "W"_"n"(1) = "T"_"n" are the signed tangent numbers 1,1,0,-2,0,16,0,-272,....
• "W"_"n"-1(1) "n" / (4^"n" − 2^"n") = "B"_"n" gives for "n" > 1 the Bernoulli numbers.
• W_"n"(1) 2^"n"("n"+1) = 1, 1, 0, 1, 0, 3,... the Genocchi numbers.
• | "W"_"n"( ["n" odd] ) | = 1,1,1,2,5,16,61,272,..., the number of alternating permutations (sometimes called up-down numbers).
• | "W"_"n"( ["n" odd] ) / "n"! | = "S"_"n"+1 the Euler Zeta numbers for "n" 0. (The expression ["n" odd] is 1 if "n" is odd, 0 otherwise).
• 2^"n" "W"_"n"(1/2) = 1,1,-3,-11,57,361,... are the generalized Euler numbers, or Springer numbers, with a different sign convention. See sequence A001586 in the Encyclopedia of Integer Sequences.

The sinusoidal character of the polynomials

The scaled Euler-Worpitzky-Chen polynomials are defined as

:$omega\_n(x)\; =\; W\_n(x),/,|,W\_n(\; [n\; operatorname\{odd\}]\; ),|\; .$

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

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

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

Diagrams of scaled Euler-Worpitzky-Chen polynomials

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] .