An asymptotic distribution theory for Eulerian recurrences with applications. (English) Zbl 1440.05026
Summary: We study linear recurrences of Eulerian type of the form
\[P_n(v) = (\alpha(v) n + \gamma(v)) P_{n - 1}(v) + \beta(v)(1 - v) P_{n - 1}^\prime(v)\qquad (n \geqslant 1),\]
with \(P_0(v)\) given, where \(\alpha(v)\), \(\beta(v)\) and \(\gamma(v)\) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of \(P_n(v)\) for large \(n\) using the method of moments and analytic combinatorial tools under varying \(\alpha(v)\), \(\beta(v)\) and \(\gamma(v)\), and apply our results to more than two hundred of concrete examples when \(\beta(v) \neq 0\) and more than three hundred when \(\beta(v) = 0\) that we gathered from the literature and from Sloane’s OEIS database. The limit laws and the convergence rates we worked out are almost all new and include normal, half-normal, Rayleigh, beta, Poisson, negative binomial, Mittag-Leffler, Bernoulli, etc., showing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used.
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05A05 | Permutations, words, matrices |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05E16 | Combinatorial aspects of groups and algebras |
| 11B83 | Special sequences and polynomials |
| 30C10 | Polynomials and rational functions of one complex variable |
| 30E15 | Asymptotic representations in the complex plane |
| 60F05 | Central limit and other weak theorems |
Keywords:
Eulerian numbers; Eulerian polynomials; recurrence relations; generating functions; limit theorems; Berry-Esseen bound; partial differential equations; singularity analysis; quasi-powers approximation; permutation statistics; derivative polynomials; asymptotic normality; singularity analysis; method of moments; Mittag-Leffler function; beta distributionSoftware:
OEISOnline Encyclopedia of Integer Sequences:
Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.Triangle of Eulerian numbers T(n,k), 0 <= k <= n, read by rows.
Triangle of 2-Eulerian numbers.
Euler’s triangle: triangle of Eulerian numbers T(n,k) (n>=0, 0 <= k <= n) read by rows.
Triangle read by rows: T(n,k) is the number of single loop solutions formed by n proper arches (connecting an odd and even vertice from 1 to 2n) above the x axis, k arches above the x axis connecting an odd vertice to a higher even vertice and a rainbow of n arches below the x axis.
Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
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