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Several polynomials associated with the harmonic numbers. (English) Zbl 1130.11011
For nonnegative integers $r$ and $n$ let $H_n^{(r)}=\sum_{1\le n_0+\cdots+n_r\le n}{{1}\over {n_0n_1\cdots n_r}}$ be the $n$th generalized harmonic number of rank $r$. In this paper, the authors develop polynomials $H_n^{(r)}(z)$ of degree $n-r$ in the complex variable $z$ generalizing the above harmonic numbers. These polynomials are given by $$ {{[-\ln(1-t)]^{1+r}}\over {t(1-t)^{1-z}}}=\sum_{n=0}^{\infty} H_n^{(r)}(z) t^n. $$ The harmonic polynomials can be expressed in terms of the generalized harmonic numbers as $$ H_n^{(r)}(z)=\sum_{k=0}^{n-r} (-1)^k H_{n}^{(r+k)}{{z^k}\over {k!}}, $$ 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 $$ {{[x-z+1]_n}\over {n!}}=\sum_{k=0}^n {{1}\over {(k+1)!}} H_n^{(k)}(z-x+1), $$ where, as usual, $[x]_n=x(x+1)\cdots (x+n-1)$. The proofs make strong use of the summation property of Riordan arrays [see {\it L. W. Shapiro, S. Getu, W.-J. Woan} and {\it L. C. Woodson}, Discrete Appl. Math. 34, No. 1--3, 229--239 (1991; Zbl 0754.05010)].

11B68Bernoulli and Euler numbers and polynomials
11B73Bell and Stirling numbers
05A10Combinatorial functions
05A15Exact enumeration problems, generating functions
Full Text: DOI
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