Recursive Harmonic Numbers and Binomial Coefficients

Aung Phone Maw

Recursive Harmonic Numbers and Binomial Coefficients

Aung Phone Maw

2017, arXiv (Cornell University)

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We define recursive harmonic numbers as a generalization of harmonic numbers. The table of recursive harmonic numbers, which is like Pascal's triangle, is constructed. A formula for recursive harmonic numbers containing binomial coefficients is also presented.

Recursive Harmonic Numbers and Binomial Coefficients Aung Phone Maw and Aung Kyaw Department of Mathematics University of Yangon Yangon, 11041, Myanmar [email protected] Abstract. We define recursive harmonic numbers as a generalization of harmonic numbers. The table of recursive harmonic numbers, which is like Pascal’s triangle, is constructed. A formula for recursive harmonic numbers containing binomial coefficients is also presented. 1. Recursive Harmonic Numbers n 1 For a positive integer n, a harmonic number Hn is defined as H n = ∑ . Here we define k =1 k th (m) m recursive harmonic number H n as follows: n 1 H n(0) = 1 ; H n( m ) = ∑ H k( m −1) for any positive integer m. k =1 k n n n 1 1 1 Since H= = ∑ = ⋅1 ∑ H k(0) = H n(1) , one can see that mth recursive harmonic ∑ n k k k = k 1= k 1 = k 1 number is a generalization of a harmonic number. 2. Table of Recursive Harmonic Numbers From the definition of mth recursive harmonic number, H n(0) = 1 and H1( m ) = 1 . For every n ≥ 2 we have n 1 H n( m ) = ∑ H k( m −1) k =1 k n −1 1 1 = ∑ H k( m −1) + H n( m −1) n k =1 k 1 (m) H= H n( m−1) + H n( m −1) n n From these facts we can construct the table of recursive harmonic numbers like Pascal’s triangle as follows: m 0 1 2 3 4 1 1 2 1 3 1 4 1 1 3 2 11 6 25 12 1 7 4 85 36 415 144 1 15 8 575 216 5845 1728 1 31 16 3661 1296 76111 20736 n 2 3. Recursive Harmonic Numbers and Binomial Coefficients A well-known formula for harmonic numbers containing binomial coefficients is n 1 n (1) = H H= (−1) k +1   . ∑ n n k k  k =1 We will show that n 1 n m) H n(= (−1) k +1 m   . ∑ k k  k =1 Proof. We will prove by induction. 1 1 1 When n = 1 , H 1( m ) = 1 = ∑ (−1) k +1 m   . k k  k =1 When m = 0 , H n( 0 ) = 1 and n 1 n n n n n =   −   +   −   +  + (−1) n +1  =  1. 0   k k  1  2  3  4 k =1 n n 1 n (0) Therefore H= (−1) k +1 0   . ∑ n k k  k =1 n ∑ (−1) k +1 By assuming the formula is true for H n( m −1) and H n( m−1) , we will show that the 1 n (m) formula is true for H n( m ) , n ≥ 2, m ≥ 1 . Since H= H n( m−1) + H n( m −1) , then n 1 ( m −1) Hn n n −1 1 n 1 n k +1 1  n − 1  ( 1) (−1) k +1 m −1   = − + ∑ ∑  m  k  k  nk1 k k  k 1= = (m) H= H n( m−1) + n = ∑ (−1)k +1 k = n −1 1 n − k n 1 1 n n +1 ( 1) 1 (−1) k +1 + − ⋅ + ∑    + m m −1 k n k  n⋅n n ⋅ k m −1  k  k 1 1= n −1 n −1 ∑ (−1) k +1 k =1 = n −1 ∑ (−1) k +1 k =1 m) H n(= n ∑ (−1) k =1 k +1 1 (n − k ) + k  n  n +1 1   + (−1) m nm k n k  1 n 1 + (−1) n +1 m m   k k  n 1 n   km k  Other formulas involving harmonic numbers and binomial coefficients can be found in [1, 2, 3] and others. References 1. J. Choi, Finite Summation Formulas involving Binomial Coefficients, Harmonic Numbers and Generalized Harmonic Numbers, J. Inequal. Appl. 2013:49 3 2. W. Chu and Q. Yan, Combinatorial Identities on Binomial Coefficients and Harmonic Numbers, Util. Math. 75 (2008) 51-66 3. M.J. Kronenburg, Some Combinatorial Identities some of which involving Harmonic Numbers, arXiv: 1103.1268v3 [math.CO] 12 Jan 2017

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