IJPAM: Volume 82, No. 4 (2013)

ON CONVOLUTION OF SOME TYPE OF THE NUMBERS
CONNECTED WITH GENERALIZED REPUNITS

Pavel Trojovský
University of Hradec Králové
Faculty of Science
Department of Mathematics
62, Rokitanského, 50003, Hradec Králové, CZECH REPUBLIC


Abstract. We will concentrate on special types of numbers

\begin{displaymath}J_{n}(k)=\sum_{i=0}^{n-2}\binom{n}{i}k^{n-2-i},\end{displaymath}

where $k$ is any nonnegative integer and $n$ is any positive integer greater than 1. These numbers are a generalization of generalized repunits $R_n(b)$. In this paper some results about divisibility of $J_{n}(k)$ are stated. Further the generating function and a $m$-fold convolution formula for the numbers $J_{n}(k)$ is found.

Received: September 28, 2012

AMS Subject Classification: 11A51, 11A07, 11Y11

Key Words and Phrases: recurrences, repunits, congruences, generating functions, convolution

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DOI: 10.12732/ijpam.v82i4.10 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 82
Issue: 4