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Eulerian polynomials
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The Eulerian polynomials were introduced by Leonhard Euler in his Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques in 1749 (first printed in 1765) where he describes a method of computing values of the zeta function at negative integers by a precursor of Abel's theorem applied to a divergent series.
Definition
The Eulerian polynomials are defined by the exponential generating function
The Eulerian polynomials can be computed by the recurrence
An equivalent way to write this definition is to set the Eulerian polynomials inductively by
The definition given is used by major authors like D. E. Knuth, D. Foata and F. Hirzebruch. In the older literature (for example in L. Comtet, Advanced Combinatorics) a slightly different definition is used, namely
The sequence of Eulerian polynomials has ordinary generating function given by the continued fraction
Three identities
An identity due to Euler is
For instance we get for
Let denote the Stirling numbers of the second kind. Frobenius proved that the Eulerian polynomials are equal to:
The third identity is called Worpitzky's identity
Here denotes the coefficient of in .
Eulerian numbers
 Main article page: Eulerian numbers
The coefficients of the Eulerian polynomials are the Eulerian numbers ,^{[1]}
This definition of the Eulerian numbers agrees with the combinatorial definition in the DLMF.^{[2]} The triangle of Eulerian numbers is also called Euler's triangle.^{[3]}



Euler's definition A(n, k) is A173018. The main entry for the Eulerian numbers in the OEIS database is A008292. It enumerates like C(n, k) albeit restricted to n ≥ 1 and k ≥ 1.
A combinatorial interpretation
Let S_{n} denote the set of all bijections (onetoone and onto functions) from {1, 2, …, n} to itself, call an element of S_{n} a permutation p and identify it with the ordered list p_{1} p_{2} … p_{n}.
Using the Iverson bracket notation [.] the number of ascents of p is defined as
where p_{n+1} ← 0. The combinatorial interpretation of the Eulerian polynomials is then given by
The table below illustrates this representation for the case n = 4.
p  asc  p  asc  p  asc  p  asc 
4321  0  4231  1  2413  2  1423  2 
3214  1  2431  1  2134  2  1342  2 
3241  1  4312  1  2314  2  4123  2 
3421  1  3142  1  2341  2  1324  2 
4213  1  4132  1  3124  2  1243  2 
2143  1  1432  1  3412  2  1234  3 
The number of permutations of {1, 2, …, 2n} with n ascents (the central Eulerian numbers) are listed in A180056.
History
Leonhard Euler introduced the polynomials in 1749 ^{[4]} in the form
Euler introduced the Eulerian polynomials in an attempt to evaluate the Dirichlet eta function
at s = 1, 2, 3,... . This led him to conjecture the functional equation of the eta function (which immediately implies the functional equation of the zeta function). Most simply put, the relation Euler was after was
Though Euler's reasoning was not rigorous by modern standards it was a milestone on the way to Riemann's proof of the functional equation of the zeta function. A short exposition of what Euler did was given by Keith Conrad on MathOverflow.
The facsimile shows Eulerian polynomials as given by Euler in his work Institutiones calculi differentialis, 1755. It is interesting to note that the original definition of Euler coincides with the definition in the DLMF.
Eulerian generating functions
We call a generating function an Eulerian generating function iff it has the form
Generating function g(t)A_{n}(t) / (1 t)^{n+1} 
n = 0  n = 1  n = 2  n = 3  n = 4  n = 5 
g(t) = 1 − t^{2}  A019590  A040000  A008574  A005897  A008511  A008512 
g(t) = 1 − t  A000007  A000012  A005408  A003215  A005917  A022521 
g(t) = t  A057427  A001477  A000290  A000578  A000583  A000584 
g(t) = 1 + t  A040000  A005408  A001844  A005898  A008514  A008515 
g(t) = 1 + t + t^{2}  A158799  A008486  A005918  A027602  A160827  A179995 
For instance the case
 g(t) = t gives the generating function of the regular orthotopic numbers, and the case
 g(t) = 1 + t gives the generating function of the centered orthotopic numbers.
The roots of the polynomials
has only (negative and simple) real roots, a result due to Frobenius. In fact the Eulerian polynomials form a Sturm sequence, that is, has n real roots separated by the roots of .
Special values of the Eulerian polynomials
x  −1/2  1/2  3/2 
2^{n}A_{n}(x)  A179929  A000629  A004123 
x  −2  −1  0 
A_{n}(x)  A087674  A155585  A000012 
x  1  2  3 
A_{n}(x)  A000142  A000670  A122704 
Assorted sequences and formulae
Let ∂r denote the denominator of a rational number r.
A122778  A_{n}(n) 
A180085  A_{n}(−n) 
A000111  A_{n}(I)(1+I)^{(1n)} ^{[5]} 
A006519  ∂(A_{n}(−1) / 2^{n}) 
A001511  log_{2}(∂(A_{2n+1}(−1) / 2^{2n+1})) 
Eulerian polynomials A_{n}(x) and Euler polynomials E_{n}(x) have a sequence of values in common (up to a binary shift). Let B_{n}(x) denote the Bernoulli polynomials and ζ(n) the Riemann Zeta function. denotes the Stirling numbers of the second kind. The formulas below show how rich in content the Eulerian polynomials are.
A155585 for all n ≥ 0 
The connection with the polylogarithm
Eulerian polynomials are related to the polylogarithm
For nonpositive integer values of s, the polylogarithm is a rational function. The first few are
A plot of these functions in the complex plane is given in the gallery^{[6]} below.
In general the explicit formula for nonpositive integer s is
See also DLMF and the section on series representations of the polylogarithm on Wikipedia. However, note that the conventions on Wikipedia do not conform to the DLMF definition of the Eulerian polynomials.
Program
(Maple) a := proc(n,m) local k; add((1)^k*binomial(n+1,k)*(m+1k)^n,k=0..m) end: A := proc(n,x) local k; `if`(n=0,1,add(a(n,k)*x^k,k=0..n1)) end: (Sage) def a(n, m) : return sum((1)^k*binomial(n+1,k)*(m+1k)^n for k in (0..m)) def A(n, x) : if n == 0 : return 1 return sum(a(n,k)*x^k for k in (0..n1))
Notes
 ↑ The Eulerian number A(n, k) is not to be confused with the value of the n^{th} Eulerian polynomial at k. For instance A(n, n) = 1,0,0,0,... whereas A_{n}(n) is A122778.
 ↑ Digital Library of Mathematical Functions, National Institute of Standards and Technology, Table 26.14.1
 ↑ The name Euler's triangle is used, for example, in Concrete Mathematics, Table 254. A virtue of this name is that it might evoke an association to Pascal's triangle, with which it shares the symmetry between left and right.
 ↑ Euler read his paper in the Königlichen Akademie der Wissenschaften zu Berlin in the year 1749 ("Lu en 1749"). It was published only much later in 1768.
 ↑ Comment from Peter Bala. In terms of the polylogarithm this is I^(n+1) 2 Li_{n}(I) = A000111(n).
 ↑ Author of the plots of the polylogarithm functions in the complex plane: Jan Homann. Public domain.
References
 P. Barry, Eulerian Polynomials as Moments, via Exponential Riordan Arrays, Journal of Integer Sequences, Vol. 14 (2011), Article 11.9.5.
 K. Conrad, Answer to: Historical question in analytic number theory, MathOverflow, Jan 29 2010.
 Leonhard Euler, E352 (Eneström Index), Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques, 1768., The Euler Archive.
 Leonhard Euler, E212 (Eneström Index), Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum (Foundations of Differential Calculus, with Applications to Finite Analysis and Series), 1755., The Euler Archive.
 Dominique Foata and MarcelPaul Schützenberger, Théorie Géométrique des Polynômes Eulériens, 1970, arXiv:math.CO/0508232.
 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 1989.
 Friedrich Hirzebruch, Eulerian polynomials, Münster J. of Math. 1 (2008), 9–14.
 Dominique Foata, Eulerian Polynomials: from Euler's Time to the Present. Invited address at the 10th Annual Ulam Colloquium, University of Florida, February 18, 2008.
See also
 P. Luschny, Generalized Eulerian Polynomials.