Geodesic
Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences
Prikladnaâ diskretnaâ matematika, no. 1 (2022), pp. 5-13

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In this paper, using numbers of a special kind ${H_{n}^{(r)}= \sum\limits_{m=r}^{n}{\ldots}\sum\limits_{l=3}^{s-1}\sum\limits_{j=2}^{l-1}\sum\limits_{i=1}^{j-1}{\dfrac{1}{ijl\ldots m}}}$, $r,n \in\mathbb{N}$, called multiharmonic numbers, incomplete closed forms of two fundamental sequences of integers given as a recursion are synthesized. The first recursion $u_{k+1}^{(m)}=(k+m)[2u_{k}^{(m)}-(k-1)u_{k-1}^{(m)}]$, ${u_{k}\in\mathbb{Z}}$, ${k\in\mathbb{N}}$, ${m\in\mathbb{Z}^{+}}$, under the conditions ${m=0}$, $u_{0}^{(0)}=u_{1}^{(0)}=1$ is factorial-generating: $u_{k}^{(0)}=k!$. The second recursion defines a sequence of Stirling numbers of the first kind ${s(n,k)}$, ${n,k\in\mathbb{Z}^{+}}$, and by the property ${|s(n,1)|=(n-1)!}$ is also factorial-generating. The resulting closed form for the first recursion is ${u_{k}^{(m)}=\sum\limits_{i=0}^{k-1}{\text{C}_{k-1}^{i}{\text{A}_{k+m-1}^{k-i}{m^{i-1}}}}}$, ${k,m\in\mathbb{N}}$, ${\text{A}_{n}^{m}}={n!}/{(n-m)!}$, ${\text{C}_{n}^{m}}={n!}/{(n-m)!m!}$. The closed form for the second recursion is ${s(n,k)= H_{n-1}^{(k-1)}{(n-1)!}{(-1)^{n+k}}}$, ${k,n\in\mathbb{N}}$. Closed forms are not complete, since they are not used for cases: ${m=k=0}$, ${n=k=0}$.
Keywords: closed forms of recurrent equations with nonlinear coefficients, interpolation of recurrent sequences, generating recursion functions, factorial-generating sequences, hyperharmonic numbers, multiharmonic numbers, Stirling numbers of the first kind. 
I. V. Statsenko. Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences. Prikladnaâ diskretnaâ matematika, no. 1 (2022), pp. 5-13. http://geodesic.mathdoc.fr/item/PDM_2022_1_a0/
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