arXiv subjects
Publications and source records attributed to Mario Catalani.
This note is dedicated to Professor Gould. The aim is to show how the identities in his book "Combinatorial Identities" can be used to obtain identities for Fibonacci and Lucas polynomials. In turn these identities allow to derive a wealth of numerical identities for Fibonacci and Lucas numbers.
We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of combinatorial identities can be obtained for selected values of the variables.
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in most cases generalize known results.
In this note we study some sequences whose ratio converges to the square root of rationals. Further we analyze some related sequences obtained when the above mentioned ratio simplifies.
We consider sequences of the type $A_n=6A_{n-1}-A_{n-2}, A_0=r, A_1=s$ ($r$ and $s$ integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type.
The problem we are dealing with is the following: find two sequences $a_n$ and $b_n$ such that the average of the first $b_n$ triangular numbers (starting with the triangular number 1) is still a triangular number, precisely the $a_n$-th triangular number. We get also some side results: for instance one of the sequence instrumental to finding the asked for sequences turns out to be a bisection of the sequence of the numerators of continued fraction convergents to $\sqrt{3}$.
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the definition of reflected sequence and inverted sequence and we establish some relationship between the coefficients of the Cayley-Hamilton equation for these matrices and the introduced sequences.
We propose two algorithms for sampling from two gamma variates possessing a negative correlation. The case of positive correlation is easily solved, so we just mention it. The main problem is the lowest value of the correlation coefficient that can be reached. The starting point of both algorithms is generation from a bivariate density with uniform negatively correlated marginals. Actually the first method uses a degenerate bivariate density since it considers two uniforms related by a linear relationship. Then we resort essentially to the inverse transform method. For both algorithms we stress restrictions on the parameters and rigidities.
We consider a sequence of sums of powers of the the roots of the cubic equation characterizing the Tribonacci sequences and derive its relationship with a particular Tribonacci sequence. Then we make a conjecture on the possible generalization.
Determinants and symmetric functions of the eigenvalues of matrices characterizing stochastic processes with indepedent increments. Relationships with Fibonacci numbers are derived.
We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants of the 2nd order principal minors of this matrix.
We know that the marginals in a Dirichlet distribution are beta variates exhibiting a negative correlation. But we can construct two linear combinations of such marginals in such a way to obtain a positive correlation. We discuss the restrictions that are to be imposed on the parameters to accomplish such a result. In the case the sampling from the Dirichlet distribution is performed through a generalization of Johnk's method we discuss the efficiency of the algorithm implementing the method.
We know that the marginals in a multinomial distribution are binomial variates exhibiting a negative correlation. But we can construct two linear combinations of such marginals in such a way to obtain a positive correlation. We discuss the restrictions that are to be imposed on the parameters of the given marginals to accomplish such a result. Next we discuss the regression function, showing that it is a linear function but not homoscedastic.
We study a) the limit of the ratio of two consecutive terms in such a sequence and b) the limit of the ratio of two terms in which one has a lag equal to 2. In the general case limit a) does not exist but we have two limiting values depending on the parity of the index. And these limits depend on the initial conditions. Limit b) exists and does not depend on the the initial conditions. Finally we seek the set of initial conditions for which this limit exists (that is, the two limiting values coincide) and obtain this limit.