×

The Pell sequence of order k, multinomial coefficients, and probability. (English) Zbl 0597.10010

The sequence \(\{P_ n^{(k)}\}^{\infty}_{n=0}\) \((k=2,3,...)\) is called by the authors Pell sequence of order k if \(P_ 0^{(k)}=0\), \(P_ 1^{(k)}=1\), and \[ P_ n^{(k)}=\sum^{n}_{i=1}2^{k-i} P^{(k)}_{n-i}\quad if\quad 2\leq n\leq k;\quad P_ n^{(k)}=\sum^{k}_{i=1}2^{k-i} P^{(k)}_{n-i}\quad if\quad n\geq k+1. \] It follows that \(\{P_ n^{(k)}\}^{\infty}_{n=0}\) is a generalized Pell sequence, since \(P_ n^{(2)}=P_ n\) (n\(\geq 0)\), where \(P_ n\) denotes the Pell sequence. The authors show that \[ P^{(k)}_{n+1}=\sum_{n_ 1,...,n_ k}\left( \begin{matrix} n_ 1+...+n_ k\\ n_ 1,...,n_ k\end{matrix} \right) 2^{-n+k(n_ 1+...+n_ k)},\quad n\geq 0, \] where the summation is taken over all nonnegative integers \(n_ 1,...,n_ k\) such that \(n_ 1+2n_ 2+...+kn_ k=n\). They also show that \[ P^{(k)}_{n+1}=(\frac{1+2^ k}{2})^ n\sum^{\infty}_{i=0}(-1)^ i\quad \left( \begin{matrix} n-ki\\ i\end{matrix} \right)\quad 2^{ki}(1+2^ k)^{-(k+1)i} \]
\[ - \frac{1}{2}(\frac{1+2^ k}{2})^{n-1}\sum^{\infty}_{i=0}(-1)^ i\quad \left( \begin{matrix} n-1-ki\\ i\end{matrix} \right)\quad 2^{ki}(1+2^ k)^{-(k+1)i},\quad n\geq 1. \] Denote by \(N_ k\) the number of independent Bernoulli trials with constant success probability p until the occurence of the k-th consecutive success, and assume that \(p=1/(1+2^ k)\). Under these assumptions the authors show that \[ P[N_ k=n+k]=2^ n P^{(k)}_{n+1}/(1+2^ k)^{n+k},\quad n\geq 0, \] which relates probability to the Pell sequence of order k. Generalizations of the above results have been obtained by C. Georghiou and the authors [Int. J. Math. Math. Sci. 6, 545-550 (1983; Zbl 0524.10008)].

MSC:

11B37 Recurrences

Citations:

Zbl 0524.10008
PDFBibTeX XMLCite
Full Text: EuDML