Read e-book online Combinatorial Identities for Stirling Numbers: The PDF

By Jocelyn Quaintance, Henry W. Gould

ISBN-10: 9814725269

ISBN-13: 9789814725262

This publication is a distinct paintings which supplies an in-depth exploration into the mathematical services, philosophy, and data of H W Gould. it really is written in a method that's available to the reader with uncomplicated mathematical wisdom, and but comprises fabric that may be of curiosity to the expert in enumerative combinatorics. This booklet starts off with exposition at the combinatorial and algebraic options that Professor Gould makes use of for proving binomial identities. those recommendations are then utilized to improve formulation which relate Stirling numbers of the second one variety to Stirling numbers of the 1st type. Professor Gould's concepts additionally offer connections among either kinds of Stirling numbers and Bernoulli numbers. Professor Gould believes his examine luck comes from his instinct on tips to notice combinatorial identities.This booklet will entice a large viewers and should be used both as lecture notes for a starting graduate point combinatorics type, or as a learn complement for the professional in enumerative combinatorics.

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Extra info for Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H W Gould

Example text

Without loss of generality assume that n ≥ 1. 26) can be rewritten as x−1 2 (−1)n 22n = n = 2n n! n−1 (2j + 1 − x) j=0 2n (1 − x)(3 − x) . . (2n − 1 − x). n! 27) is zero for all x ∈ {1, 3, 5, 7, . . , 2n − 1} = {2m + 1}m=0 . 26) is zero for these same n values. Let x = 2m + 1 for 0 ≤ m ≤ n − 1. 26) 2n−2m−1 becomes nj=0 (−1)j 2m+1 . If 2m + 1 > n, we can extend the j n−j range of summation to 2m + 1 since 2n−2m−1 vanishes whenever j > n. n−j If 2m + 1 < n, we can truncate the range of summation at 2m + 1 since 2m+1 = 0 whenever j > 2m + 1.

On the other hand 2 (−1)n 22n = = n−1 2 n = 24m 4m 2 2m−1 2 2m−1 2 24m 2m−1 2 2m−3 2 2m−1 2 2m − 1 ... 2m−1 2 . . 12 − 21 (2m)! −m+1 (2m)! − 32 . . 2m−1 2 − m ... 2m−1 2 −2m+1 2 = 22m (2m − 1)(2m − 3) . . (1)(−1)(−3) . . (−2m + 1) (2m − 1)(2m − 3) . . (3)(1)(2m)(2m − 2) . . (4)(2) = (−1)m 2m (1)(3) . . (2m − 1) m! 22m (−1)(−3) . . (−2m + 1) = · 2m m! m! m! = (−1)m (2m)! 2m = (−1)m . m!

26) can be rewritten as x−1 2 (−1)n 22n = n = 2n n! n−1 (2j + 1 − x) j=0 2n (1 − x)(3 − x) . . (2n − 1 − x). n! 27) is zero for all x ∈ {1, 3, 5, 7, . . , 2n − 1} = {2m + 1}m=0 . 26) is zero for these same n values. Let x = 2m + 1 for 0 ≤ m ≤ n − 1. 26) 2n−2m−1 becomes nj=0 (−1)j 2m+1 . If 2m + 1 > n, we can extend the j n−j range of summation to 2m + 1 since 2n−2m−1 vanishes whenever j > n. n−j If 2m + 1 < n, we can truncate the range of summation at 2m + 1 since 2m+1 = 0 whenever j > 2m + 1.

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Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H W Gould by Jocelyn Quaintance, Henry W. Gould


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