
EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLINGEULERLAHBELL
 Author(s):
 Leon Chang, Hungping 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/g1bqgm07
 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)^(n1)▒〖2^k 〈■(n@k)〉 〗=∑_(k=1)^n▒(∑_(j1)^(k+1)▒[■(k+1@j)] ){■(n@k)} , the righthand 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(n1), W(3)=A(k), W(4)=A(n+k1) +2A(1)A(2), W(5)=A(nk+1), W(6)=A(k+2)2A(1) and W(7)=A(2nk) in the recursive formula T(n, k)=W(u)T(n1,k1)+W(v)T(n1,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+(i1)d  1;2), [■(n@k)]_(1;1)= [■(n@k)], possessing diagonal (Stirling) polynomials; {■(n@k)}_(a;d)=T(n, k  A(i)=a+(i1)d  1;3), {■(n@k)}_(1;1)={■(n@k)}, possessing diagonal polynomials; Lad(n, k)=T(n, k  A(i)=a+(i1)d  1;4), L11(n, k)=L(n, k), possessing diagonal polynomials. 2. Eulerian: u>1, 1≤k≤n1 and T(n, 0), n > 0, varies for each array. 〈■(n@k)〉_(a;d)=T(n, k+1  A(i)=a+(i1)d  5;6), T(n, 0)=[A(2)2A(1)]T(n1, 0) for n > 0; 〈〈■(n@k)〉 〉_(a;d)=T(n, k+1  A(i)=a+(i1)d  7;6), T(n, 0)=[A(2)2A(1)]T(n1, 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, qGaussian coefficient, Bell number, Ordered Bell polynomial, Python.
 Metadata:
 xml
 Published as:
 Book chapter Show details
 Publisher:
 Lenox Institute Press, Newtonville, NY, 121280405, USA.
 Pub. Date:
 January 2021
 Book Title:
 \"Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)\"
 Author/Editor:
 Wang, Lawrence K. 王抗曝 and Tsao, Hungping 曹恆平 (editors)
 Chapter:
 1
 Page Range:
 1  99
 ISBN:
 9780989087032
 Status:
 Published
 Last Updated:
 2 years ago
 License:
 All Rights Reserved
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EVOLUTIONARY MATHEMATICS AND SCIENCE FOR GENERAL FAMOUS NUMBERS: STIRLINGEULERLAHBELL