Harmonic Number Expansions of the Ramanujan Type
Sign Up to like & getrecommendations! Published in 2018 at "Results in Mathematics"
DOI: 10.1007/s00025-018-0920-8
Abstract: In this paper, we establish three (general) asymptotic expansions of the Ramanujan type for the harmonic numbers, and give the corresponding recurrences of the coefficient sequence or parameter sequences in these expansions. We also present… read more here.
Keywords: ramanujan type; formula; number expansions; harmonic number ... See more keywords
On the Ramanujan Harmonic Number Expansion
Sign Up to like & getrecommendations! Published in 2018 at "Results in Mathematics"
DOI: 10.1007/s00025-018-0925-3
Abstract: In this paper, we give a recursive relation for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number, without the Bernoulli numbers and polynomials. read more here.
Keywords: number expansion; harmonic number; ramanujan harmonic; expansion ... See more keywords
Generalized harmonic number sums and quasisymmetric functions
Sign Up to like & getrecommendations! Published in 2020 at "Rocky Mountain Journal of Mathematics"
DOI: 10.1216/rmj.2020.50.1253
Abstract: We express some general type of infinite series such as $$ \sum^\infty_{n=1}\frac{F(H_n^{(m)}(z),H_n^{(2m)}(z),\ldots,H_n^{(\ell m)}(z))} {(n+z)^{s_1}(n+1+z)^{s_2}\cdots (n+k-1+z)^{s_k}}, $$ where $F(x_1,\ldots,x_\ell)\in\mathbb Q[x_1,\ldots,x_\ell]$, $H_n^{(m)}(z)=\sum^n_{j=1}1/(j+z)^m$, $z\in (-1,0]$, and $s_1,\ldots,s_k$ are nonnegative integers with $s_1+\cdots+s_k\geq 2$, as a linear combination of… read more here.
Keywords: sums quasisymmetric; number sums; ldots ell; quasisymmetric functions ... See more keywords
Harmonic number identities via polynomials with r-Lah coefficients
Sign Up to like & getrecommendations! Published in 2020 at "Comptes Rendus Mathematique"
DOI: 10.5802/crmath.53
Abstract: In this paper, polynomials whose coefficients involve r -Lah numbers are used to evaluate several summation formulae involving binomial coefficients, Stirling numbers, harmonic or hyperharmonic numbers. Moreover, skew-hyperharmonic number is introduced and its basic properties… read more here.
Keywords: number; number identities; harmonic number; via polynomials ... See more keywords