Application of multiharmonic numbers for the synthesis of closed forms of parametrically modified factorial generating sequences

I. V. Statsenko

Prikladnaâ diskretnaâ matematika, no. 1 (2022), pp. 5-13

Résumé

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: ${m=k=0}$, ${n=k=0}$.

Détail

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.