Abstract
In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these connections, we reach several identities for harmonic and hyperharmonic numbers.
Highlights
The nth harmonic number Hn is defined by n1Hn = k=1 k, with the assumption H0 = 0
There is an enormous literature about the identities involving harmonic numbers with binomial coefficients
Stirling numbers and Bernoulli numbers: Benjamin and Quinn [5, Identity 4] used combinatorial technique to obtain nn n+1 k=1 k k = n!Hn = 2, where n k is the Stirling number of the first kind
Summary
Spivey [48] exhibited many combinatorial sums, for instance, the binomial harmonic identity n n (−1)k+1. There exist many elegant identities involving hyperharmonic numbers Some of these identities are exhibited by using combinatorial technique [4], Euler–Siedel matrix [24, 39], derivative and difference operators [22, 23, 26, 38], Pascal type matrix [15]. One of the generalizations is Euler sum of hyperharmonic numbers which has been evaluated in terms of zeta functions [6, 21, 25, 40, 50]. Properties of the arising polynomials give rise to binomial hyperharmonic identity and several summation formulas involving (hyper)harmonic numbers. Several new formulas for the r -Lah numbers are presented
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