An asymptotic distribution theory for Eulerian recurrences with applications Hsien-Kuei Hwanga,1,∗, Hua-Huai Chernb,2, Guan-Huei Duhc aInstitute of Statistical Science, Academia Sinica, Taipei 115, Taiwan bDepartment of Computer Science, National Taiwan Ocean University, Keelung 202, Taiwan cInstitute of Statistical Science, Academia Sinica, Taipei 115, Taiwan Abstract We study linear recurrences of Eulerian type of the form 0 Pn(v) = (α(v)n + γ(v))Pn−1(v) + β(v)(1 − v)Pn−1(v)(n > 1); with P0(v) given, where α(v); β(v) and γ(v) are in most cases polynomials of low degrees. We characterize the various limit laws of the coefficients of Pn(v) for large n using the method of moments and analytic combinatorial tools under varying α(v); β(v) and γ(v), and apply our results to more than two hundred of concrete examples when β(v) , 0 and more than three hundred when β(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., show- ing the surprising richness and diversity of such a simple framework, as well as the power of the approaches used. 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 distribution. Contents 1 Introduction4 2 A normal limit theorem 10 arXiv:1807.01412v2 [math.CO] 3 Nov 2019 2.1 Mean value of Xn ................................. 11 2.2 Recurrence relation for higher central moments . 13 2.3 Asymptotics of V(Xn)............................... 15 ∗Corresponding author Email addresses: [email protected] (Hsien-Kuei Hwang), [email protected] (Hua-Huai Chern), [email protected] (Guan-Huei Duh) 1Partially supported by an Investigator Award from Academia Sinica under the Grant AS-IA-104-M03. 2Partially supported by Ministry of Science and Technology (Taiwan) under the Grant MOST-106-2115-M- 019-001. Preprint submitted to Elsevier November 5, 2019 2.4 Asymptotics of higher central moments . 15 2.5 Mean and variance in a more general setting . 16 3 A complex-analytic approach 16 3.1 The partial differential equation and its resolution . 17 3.2 Singularity analysis and quasi-powers theorem for CLT . 19 ) 1 1 4 Applications I: (α(v); β(v)) = (qv; qv) = N 2 n; 12 n 22 4.1 The class A (p; q; r)................................ 23 4.2 q = 1........................................ 24 4.3 q = 2........................................ 27 4.4 General q > 0................................... 29 4.5 Other extensions with the same CLT and their variants . 31 4.5.1 The two examples in the Introduction . 32 4.5.2 r-Eulerian numbers again . 32 4.5.3 Eulerian numbers of type D ....................... 33 4.5.4 Exponential perturbation . 33 4.5.5 Eulerian polynomials multiplied by 1 + v ................ 34 5 Applications II: α(v) , β(v) or quadratic α(v), β(v) 34 2 ) q q 5.1 Polynomials with (α(v); β(v)) = (qv; v) = N q+1 n; (q+1)2(q+2) n ........ 34 1 1 2 − 1 ) 2 5.1.1 q = 2 = N 3 n; 45 n; n ........................ 36 1 1 − 1 ) 2 5.1.2 q = 1 = N 2 n; 12 n; n ........................ 37 2 1 − 1 ) 2 5.1.3 q = 2 = N 3 n; 9 n; n ........................ 37 3 9 − 1 ) 2 5.1.4 q = 3 = N 4 n; 80 n; n ........................ 38 2 ) q q 5.1.5 q > 1 = N q+1 n; (q+1)2(q+2) n ..................... 39 5.2 Polynomials with (α(v); β(v)) = (p + qv; v) ) q q(p+1)(p+q) = N p+q+1 n; (p+q+1)2(p+q+2) n .......................... 39 1 1 ) 1 23 5.3 Polynomials with (α(v); β(v)) = 2 (1 + v); 2 (3 + v) = N 6 n; 180 n ...... 41 5.4 Polynomials with quadratic α(v)......................... 42 2 ) 2 8 5.4.1 (α(v); β(v)) = (v ; v(1 + v)) = N 3 n; 45 n ............... 42 1 2 1 2 ) 1 5 5.4.2 (α(v); β(v)) = 2 (1 + v ); 2 (1 + v ) = N 2 n; 12 n ........... 44 ) 3 7 5.4.3 (α(v); β(v)) = (v(1 + v); v(1 + v)) = N 4 n; 48 n ............ 45 2 ) 1 5.4.4 (α(v); β(v)) = (2v ; v(1 + v)) = N n; 3 n ............... 45 5.5 Polynomials with an extra normalizing factor . 46 ) 1 1 5.5.1 (α(v); β(v)) = (2qv; q(1 + v)) = N 2 n; 4 n .............. 47 ) 1 3 5.5.2 (α(v); β(v)) = (2(1 + v); 3 + v) = N 4 n; 16 n ............. 48 ) 1 1 5.5.3 (α(v); β(v)) = (q(1 + 3v); 2qv) = N 2 n; 8 n .............. 49 ) 1 5 5.5.4 (α(v); β(v)) = (5 + 3v; 2(1 + v)) = N 4 n; 32 n ............. 50 1 1 ) 1 2 5.5.5 (α(v); β(v)) = 3 (7 + 2v); 3 (5 + 4v) = N 9 n; 27 n .......... 51 2 ) 1 5.5.6 (α(v); β(v)) = (1 + 3v ; v(1 + v)) = N n; 2 n ............. 51 2 − ) q+1 q −1 5.6 Polynomials with (α(v); β(v)) = ( 1 + (q + 1)v; qv) = N 2q n; 12q2 n .... 51 2 6 Non-normal limit laws 53 6.1 Recurrence for the factorial moments . 54 6.2 EGF and PDE . 56 6.3 Discrete limit laws . 57 6.4 Continuous limit laws . 58 7 Applications III: non-normal discrete limit laws 62 7.1 Poisson limit laws: β0 = 0............................. 62 7.2 Geometric and negative-binomial limit laws: β0 < 0............... 63 7.3 A Bernoulli limit law . 64 8 Applications IV: non-normal continuous limit laws 64 β 8.1 Beta limit laws and their mixtures ( α = −1) ................... 64 8.1.1 Uniform (Beta(1; 1)) limit laws . 65 1 1 8.1.2 Arcsine (Beta 2 ; 2 )law ......................... 66 8.1.3 Beta(q; q) with q > 1........................... 66 8.1.4 Beta(p; q) with p , q ........................... 66 8.1.5 Beta mixtures . 67 8.2 Uniform limit laws again . 68 β − 1 8.3 Rayleigh and half-normal limit laws ( α = 2 ).................. 69 8.3.1 Characterizations of Rayleigh and half-normal limit laws . 70 8.3.2 Examples. I. Rayleigh laws . 74 8.3.3 Examples. II. Half-normal laws . 75 8.4 Other limit laws . 76 8.4.1 Mittag-Leffler limit laws . 76 8.4.2 A mixture of discrete and continuous laws . 77 9 Extensions 77 9.1 Non-homogeneous recurrence . 78 9.2 Eulerian recurrences involving Pn−2(v)...................... 79 9.3 Systems of Eulerian recurrences . 81 9.4 Recurrences depending on parity . 82 9.5 1 − v 7! 1 − sv .................................. 83 1 1 − 1 1 − 9.5.1 From N 2 n; 12 n to N 2 log 2 n; (log 2)2 2 n ........... 83 1 1 1 1 9.5.2 From N 4 n; 16 n to N 3 n; 18 n ..................... 85 2 1 9.5.3 From N 3 n; 9 n to 2 2q − q−1 (q−1) −q(q−1−log q) − 2q N q−1 q−1−log q n; (q−1−log q)2 (q−1)2 n ............ 86 9.5.4 Non-normal limit laws . 86 9.6 Polya´ urn models . 86 0 0 9.7 Pn(v) = (α(v)n + γ(v))Pn−1(v) + β(v)(1 − v)Pn−1(v)................ 88 9.8 Extended Eulerian recurrences of Cauchy-Euler type . 89 9.9 A multivariate Eulerian recurrence . 90 10 The degenerate case: β(v) ≡ 0 =) Pn(v) = an(v)Pn−1(v) 91 11 Conclusions 95 3 Appendices 98 A Proof of (121) 98 B Some OEIS sequences satisfying Pn(v) = an(v)Pn−1(v) 99 1. Introduction The Eulerian numbers, first introduced and presented by Leonhard Euler in 1736 (and pub- lished in 1741; see [90] and [91, Art. 173–175]) in series summations, have been widely studied because of their natural occurrence in many different contexts, ranging from finite differences to combinatorial enumeration, from probability distribution to numerical analysis, from spline approximation to algorithmics, etc.; see the books [18, 101, 153, 200, 212, 221, 225] and the references therein for more information. See also the historical accounts in the papers [27, 144, 231, 238]. Among the large number of definitions and properties of the Eulerian DnE numbers k , the one on which we base our analysis is the recurrence 0 Pn(v) = (vn + 1 − v)Pn−1(v) + v(1 − v)Pn−1(v)(n > 1); (1) P DnE k with P0(v) = 1, where Pn(v) = 06k6n k v . In terms of the coefficients, this recurrence translates into * + * + * + n n − 1 n − 1 = (k + 1) + (n − k) (n; k 1); (2) k k k − 1 > DnE D0E with k = 0 for k < 0 or k > n except that 0 := 1. We extend the recurrence (1) by considering the more general Eulerian recurrence 0 Pn(v) = (α(v)n + γ(v))Pn−1(v) + β(v)(1 − v)Pn−1(v)(n > 1); (3) with P0(v), α(v); β(v) and γ(v) given (they are often but not limited to polynomials). We are concerned with the limiting distribution of the coefficients of Pn(v) for large n when the coeffi- cients are nonnegative.