General Eulerian polynomials as moments using exponential Riordan arrays. (English) Zbl 1342.11028
Summary: Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the general Eulerian polynomials, as defined by Xiong, Tsao and Hall [T. Xiong et al., J. Math. 2013, Article ID 629132, 9 p. (2013; Zbl 1268.11035)], are moment sequences for simple families of orthogonal polynomials, which we characterize in terms of their three-term recurrence. We obtain the generating functions of this polynomial sequence in terms of continued fractions, and we also calculate the Hankel transforms of the polynomial sequence. We indicate that the polynomial sequence can be characterized by the further notion of generalized Eulerian distribution first introduced by M. Morisita [Measuring of habitat value by environmental density method. In: G. P. Patil et al. (eds.), Statistical Ecology, The Pennsylvania State University Press, 379–401 (1971)]. We finish with examples of related Pascal-like triangles.
MSC:
11B68 | Bernoulli and Euler numbers and polynomials |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
Eulerian number; Eulerian polynomial; general Eulerian number; generalized Eulerian polynomial; Euler’s triangle; exponential Riordan array; orthogonal polynomial; moment; Hankel transformCitations:
Zbl 1268.11035Software:
OEISOnline Encyclopedia of Integer Sequences:
Double factorial of even numbers: (2n)!! = 2^n*n!.Expansion of e.g.f. exp(-x)/(1-2*x).
Number of necklaces of partitions of n+1 labeled beads.
Order of alternating group A_n, or number of even permutations of n letters.
Number of generalized weak orders on n points.
Number of chains in power set of n-set.
Pascal’s triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n.
Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.
Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).
a(n) = 2*n*a(n-1) + 1 with a(0) = 1.
STIRLING transform of [1,1,2,4,8,16,32,...].
T(n, k) = Sum_{j=k..n} binomial(n, j)*E1(j, j-k), where E1 are the Eulerian numbers A173018. Triangle read by rows, T(n, k) for 0 <= k <= n.
Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k).
a(n) is the number of elements in the Coxeter complex of type B_n (or C_n).
Expansion of e.g.f. exp(2x)*sech(x).
Expansion of e.g.f.: 2*exp(2*x) / (3 - exp(2*x)).