Central Limit Theorems for Combinatorial Numbers Associated with Laguerre Polynomials

Igoris Belovas

doi:10.3390/math10060865

Abstract

In this paper, we study limit theorems for numbers satisfying a class of triangular arrays, which are defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain analytical expressions for the semi-exponential generating function of several classes of the numbers, including combinatorial numbers associated with Laguerre polynomials. We apply these results to prove the numbers’ asymptotic normality and specify the convergence rate to the limiting distribution.

Highlights

In this research, we establish limit theorems for combinatorial numbers satisfying a class of triangular arrays, extending, the investigations of Canfield [1], Kyriakoussis [2], Kyriakoussis and Vamvakari [3–6], and Belovas [7]
We consider numbers, which are defined by a bivariate linear recurrence with bivariate linear coefficients
We have proved limit theorems for three categories of numbers satisfying a class of triangular arrays, which are defined by a bivariate linear recurrence with bivariate linear coefficients, including combinatorial numbers associated with Laguerre polynomials

Summary

Introduction

We establish limit theorems for combinatorial numbers satisfying a class of triangular arrays, extending, the investigations of Canfield [1], Kyriakoussis [2], Kyriakoussis and Vamvakari [3–6], and Belovas [7]. The result is used to obtain generating functions and analytic expressions for other numbers, satisfying a class of triangular arrays. The generating function F(x, y) satisfies the linear first-order partial differential equation (1 − ψ1,1xy − ψ2,1x)Fx − (ψ1,2y2 + ψ2,2y)Fy = (ξ1y + ξ2)F, (6). The partial differentiation of the double semi-exponential generating function F(x, y) at (0, 0) yields us the analytic expressions of the numbers. Under conditions of Theorem 1, the function Λ(x, y) = F(x, y)Am(y) satisfies a linear first-order homogeneous partial differential equation for ψ1,2, ψ2,2 = 0,. Note that Corollary 2 yields us the analytic expression for the ordinary Lah numbers, which are generated by the matrix. Combining (32) with the result of the Theorem 4 for Ln,k (see (24)), and substituting into (27), we receive the second statement of the corollary

Limit Theorems for Numbers Satisfying a Class of Triangular Arrays

Conclusions