Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1130.11011
Cheon, Gi-Sang; Hwang, Suk-Geun; Lee, Sang-Gu
Several polynomials associated with the harmonic numbers. (English)
[J] Discrete Appl. Math. 155, No. 18, 2573-2584 (2007). ISSN 0166-218X

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. \par 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)].
[Florian Luca (Morelia)]

MSC 2000:: *11B68 Bernoulli numbers, etc.
11B73 Bell and Stirling numbers
05A10 Combinatorial functions
05A15 Combinatorial enumeration problems

Keywords: harmonic numbers; Riordan arrays; harmonic polynomials; generalized Stirling polynomials; Nörlund polynomials

Citations: Zbl 0754.05010

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]