Extended Fibonacci numbers and polynomials with probability applications (Q1777732)

Let \(F_n^{(k)}\), \(n\geq 0\), be the Fibonacci sequence of order \(k\). The reviewer and \textit{A. A. Muwafi} [Fibonacci Q. 20, 28--32 (1982; Zbl 0476.60008)] showed that the number of arrangements of \(\{0,1\}\)-sequences of length \(n\), such that the last symbol is \(0\) and \(k\) consecutive \(1\)'s do not appear in the sequence, is \(F_{n+1}^{(k)}\). The author presently introduces the following generalization of \(F_n^{(k)}\), \(n\geq 0\). The sequence \(A_n^{(k,r)}\) \((n\geq 0\), \(k,r\geq 2)\) is said to be the sequence of extended Fibonacci numbers if its \(n\)th term is the number of arrangements of \(\{0,1\}\)-sequences of length \(n\), such that the last symbol is \(0\) and there do not exist in the sequence any \(k\) consecutive \(1\)'s (or more) and any \(r\) consecutive \(0\)'s (or more). He derives an expansion of it in terms of multinomial coefficients, its generating function, and two recurrences for it. Furthermore, he introduces and studies the sequence of extended Fibonacci polynomials \(A_n^{(k,r)}(x)\) \((n\geq 0\), \(r\geq 2)\) as an appropriate extension of \(A_n^{(k,r)}\), in the sense that \(A_n^{(k,r)}(1)= A_n^{(k ,r)}\), \(n\geq 0\). The paper ends with probability applications regarding the random variable \(W\), which denotes the waiting time (in independent Bernoulli trials with success probability \(p\) \((0< p< 1)\)) until a success run of length \(k\) or a failure run of length \(r\) occurs, whichever comes sooner. The latter are alternative to and compare well with recent results of \textit{D. L. Antzoulakos} and the reviewer in [Ann. Inst. Stat. Math. 49, No. 3, 531--539 (1997; Zbl 0947.60503)].