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Several polynomials associated with the harmonic numbers. (English) Zbl 1130.11011

For nonnegative integers $r$ and $n$ let ${H}_{n}^{\left(r\right)}={\sum }_{1\le {n}_{0}+\cdots +{n}_{r}\le n}\frac{1}{{n}_{0}{n}_{1}\cdots {n}_{r}}$ be the $n$th generalized harmonic number of rank $r$. In this paper, the authors develop polynomials ${H}_{n}^{\left(r\right)}\left(z\right)$ of degree $n-r$ in the complex variable $z$ generalizing the above harmonic numbers. These polynomials are given by

$\frac{{\left[-ln\left(1-t\right)\right]}^{1+r}}{t{\left(1-t\right)}^{1-z}}=\sum _{n=0}^{\infty }{H}_{n}^{\left(r\right)}\left(z\right){t}^{n}·$

The harmonic polynomials can be expressed in terms of the generalized harmonic numbers as

${H}_{n}^{\left(r\right)}\left(z\right)=\sum _{k=0}^{n-r}{\left(-1\right)}^{k}{H}_{n}^{\left(r+k\right)}\frac{{z}^{k}}{k!},$

which is analogous to the formula relating Bernoulli polynomials and Bernoulli numbers.

In the paper, the authors prove various relations between the generalized harmonic polynomials and other interesting sequences of polynomials such as generalized Stirling polynomials, Bernoulli polynomials, multiple Gamma functions, Cauchy polynomials and Nörlund polynomials. For example, Theorem 5.1 shows that

$\frac{{\left[x-z+1\right]}_{n}}{n!}=\sum _{k=0}^{n}\frac{1}{\left(k+1\right)!}{H}_{n}^{\left(k\right)}\left(z-x+1\right),$

where, as usual, ${\left[x\right]}_{n}=x\left(x+1\right)\cdots \left(x+n-1\right)$. The proofs make strong use of the summation property of Riordan arrays [see L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, Discrete Appl. Math. 34, No. 1–3, 229–239 (1991; Zbl 0754.05010)].

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11B73 Bell and Stirling numbers 05A10 Combinatorial functions 05A15 Exact enumeration problems, generating functions