Algebraic and Probabilistic Aspects of Some Binary Recurrence Sequences

Abstract

Since its discovery in the year 1999, the sequence of balancing numbers (also known as the balancing sequence) has been motivating researchers to work on many algebraic and analytic properties associated with it. The sequence of Lucas-balancing numbers, which is closely entangled with the balancing numbers, also comes to picture while dealing with these properties. The solutions of certain Diophantine equations involving Lucas balancing numbers are expressible in terms of balancing, Lucas-balancing, Pell or associated Pell numbers. The balancing sequence is a strong divisibility sequence, but the Lucas balancing sequence is not a divisibility sequence. However, under some restrictions on the indices, certain subsets of balancing and Lucas-balancing numbers are divisible by the powers of balancing and Lucas-balancing numbers. The Pell and associated Pell sequences are very closely linked with the balancing sequence by multiple means. Similar divisibility properties also hold good for the Pell and associated Pell sequences. With a minor modification of the recurrence relation, the Pell numbers can be generalized to Pell polynomials. Certain products of Pell polynomials can be expressed in terms of some special orthogonal polynomials. Markov chains are a special class of stochastic processes where a probabilistic statement about any future state depends on the present state and is not affected by any additional information about the past states. The probability of passing from state i to state j in one transition is denoted by pij and is called an one step transition probability. The arrangement of these transition probabilities in a matrix is known as the transition probability matrix of the Markov chain. The probability of passing from state i to state j in n transitions is denoted by p(n) ij and is called an n-step transition probability. The n-step transition probability matrix is nothing but the n-th power of the one step transition probability matrix. As n increases, the dependence of the n-step transition probabilities p(n) ij decreases on i and pj = limn!1 p(n)ijdepends only on j and is known as a steady state transition probability. If the elements of a transition probability matrix are suitably chosen, then the numerators and denominators of the steady state probabilities can be seen to be balancing, Lucas-balancing, cobalancing, Lucas-cobalancing and balancing-like numbers. A balancing-like sequence, which generalizes the balancing sequence and the nonnegative integers sequence, is defined as xn = Axn 􀀀 xn􀀀1 with x0 = 0; x1 = 1. To avoid the viii nonnegative integers sequence, A is usually chosen to be a natural number greater than 2. But, this sequence admits further generalization, the coefficient A in the right hand side can be allowed to be a random variable with some given probability distribution having moments of all order. However, in this case, the resulting sequence will be a sequence of random polynomials in A, which is, indeed, a stochastic process, which can be better renamed as a random balancing-like sequence. Although, a balancing-like sequence is a binary recurrence sequence, the sequence of expectations may sometimes become a recurrence sequence of higher order. Moreover, if A is any random variable which is finite with probability 1, then the limit of the ratio of expectations of (n+1)-st term and n-th term remains finite; otherwise, this limit becomes infinite. It is an interesting work to study some of other differences associated with the deterministic and the random recurrence sequences.