Extended Fibonacci numbers and polynomials with probability applications
نویسنده
چکیده
The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.
منابع مشابه
On convolved generalized Fibonacci and Lucas polynomials
We define the convolved hðxÞ-Fibonacci polynomials as an extension of the classical con-volved Fibonacci numbers. Then we give some combinatorial formulas involving the hðxÞ-Fibonacci and hðxÞ-Lucas polynomials. Moreover we obtain the convolved hðxÞ-Fibo-nacci polynomials from a family of Hessenberg matrices. Fibonacci numbers and their generalizations have many interesting properties and appli...
متن کاملFibonacci numbers and orthogonal polynomials
We prove that the sequence (1/Fn+2)n≥0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = (1− √ 5)/(1+ √ 5). We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices (1/F...
متن کاملNEW CONNECTION FORMULAE BETWEEN (p, q)−FIBONACCI POLYNOMIALS AND CERTAIN JACOBI POLYNOMIALS
The main purpose of this article is to solve the connection problems between (p, q)−Fibonacci polynomials and the two polynomials, namely Chebyshev polynomials of third and fourth kinds which are considered as two nonsymmetric polynomials of the Jacobi polynomials. Moreover, the inversion connection formulae for the latter formulae are given. We show that all the connection coefficients are exp...
متن کاملModified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($alpha ,beta $)
The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...
متن کاملOn median nature and enumerative properties of Fibonacci-like cubes
Fibonacci cubes, extended Fibonacci cubes, and Lucas cubes are induced subgraphs of hypercubes 9 defined in terms of Fibonacci strings. It is shown that all these graphs are median. Several enumeration results on the number of their edges and squares are obtained. Some identities involving Fibonacci 11 and Lucas numbers are also presented. © 2005 Published by Elsevier B.V. 13
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2004 شماره
صفحات -
تاریخ انتشار 2004