• ### EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLING-EULER-LAH-BELL

Author(s):
Leon Chang, Hung-ping Tsao
Editor(s):
Lawrence K Wang (see profile)
Date:
2021
Group(s):
Digital Books
Subject(s):
Mathematics
Item Type:
Book chapter
Tag(s):
Pascal, Stirling, Eulerian, Lah, Bell
Permanent URL:
http://dx.doi.org/10.17613/g1bq-gm07
Abstract:
We first introduce Pascal, Stirling, Eulerian, Lah and Bell numbers via sorting, then generalize Stirling numbers of both kinds [■(n@k)], {■(n@k)}, Eulerian numbers of two orders 〈■(n@k)〉, 〈〈■(n@k)〉 〉, Lah numbers L(n,k)=∑_(j=1)^n▒[■(n@j)] {■(j@k)} and ∑_(k=0)^(n-1)▒〖2^k 〈■(n@k)〉 〗=∑_(k=1)^n▒(∑_(j-1)^(k+1)▒[■(k+1@j)] ){■(n@k)} , the right-hand side of which is an ordered Bell polynomial, from the natural sequence based to arithmetically progressive sequences based. We further construct two types of arrays with any infinite sequence base A(i): T(n, k | A(i) | u;v), with T(0,0)=1, and u, v each indicating which weight to be used among W(1)=1, W(2)=A(n-1), W(3)=A(k), W(4)=A(n+k-1) +2A(1)-A(2), W(5)=A(n-k+1), W(6)=A(k+2)-2A(1) and W(7)=A(2n-k) in the recursive formula T(n, k)=W(u)T(n-1,k-1)+W(v)T(n-1,k). 1. Stirling: u=1, 0≤k≤n and the initial values T(n, 0)=0 for n > 0. [■(n@k)]_(a;d)=T(n, k | A(i)=a+(i-1)d | 1;2), [■(n@k)]_(1;1)= [■(n@k)], possessing diagonal (Stirling) polynomials; {■(n@k)}_(a;d)=T(n, k | A(i)=a+(i-1)d | 1;3), {■(n@k)}_(1;1)={■(n@k)}, possessing diagonal polynomials; Lad(n, k)=T(n, k | A(i)=a+(i-1)d | 1;4), L11(n, k)=L(n, k), possessing diagonal polynomials. 2. Eulerian: u>1, -1≤k≤n-1 and T(n, 0), n > 0, varies for each array. 〈■(n@k)〉_(a;d)=T(n, k+1 | A(i)=a+(i-1)d | 5;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0; 〈〈■(n@k)〉 〉_(a;d)=T(n, k+1 | A(i)=a+(i-1)d | 7;6), T(n, 0)=[A(2)-2A(1)]T(n-1, 0) for n > 0. We shall introduce more Eulerian arrays with various initial values T(n, 0) for n > 0 and prove the existence of the general Stirling polynomials. Python programs are used to produce tables for both types of arrays along with difference tables of their diagonals to facilitate the calculation of diagonal polynomials in the case of existence.
Notes:
KEYWORDS: Natural sequence, Binomial coefficient, Stirling number, Lah number, Natural sequence, Pascal triangle, Bernoulli coefficient, Arithmetically progressive sequence, Recursive formula, Sorting, Cycle, Subset, Eulerian number, Second generation Stirling numbers, Second generation Eulerian numbers, Stirling polynomial, Diagonal polynomial, q-Gaussian coefficient, Bell number, Ordered Bell polynomial, Python.
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