A multi-parameter generalization of the symmetric algorithm

2018 ◽  
Vol 68 (4) ◽  
pp. 699-712
Author(s):  
José L. Ramírez ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Fibonacci Numbers ◽  
Recursive Formula ◽  
Binomial Coefficient ◽  
Fibonacci Polynomials ◽  
Combinatorial Argument ◽  
Recurrent Sequences ◽  
Seidel Method ◽  
Multivariate Generalization

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Qing Zou ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Chebyshev Polynomials ◽  
Triangular Matrix ◽  
Higher Order ◽  
Catalan Numbers ◽  
Lower Triangular Matrix ◽  
Fibonacci Polynomials ◽  
Delannoy Numbers ◽  
Lower Triangular

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

scholarly journals Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants

2017 ◽  
Author(s):  
Feng Qi ◽  
Jing-Lin Wang ◽  
Bai-Ni Guo
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Closed Expression

In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

A note on the time-dependent and the stationary behaviour of a semi-infinite reservoir subject to a combination of Markovian inflows

1971 ◽  
Vol 8 (04) ◽  
pp. 708-715 ◽  
Author(s):  
Emlyn H. Lloyd
Keyword(s):  
Functional Equation ◽  
Explicit Formula ◽  
Generating Function ◽  
Present Theory ◽  
The Other ◽  
Time Dependent ◽  
Independent Sequence ◽  
Time Dependent Solution ◽  
Infinite Reservoir ◽  
Stationary Behaviour

The present theory of finite reservoirs is not rich in general theorems even when of the simple Moran type, with unit draft and stationary discrete independent-sequence inflows. For the corresponding systems with unbounded capacity however there are two classes of results which have been known for some time. One of these classes is concerned with the time-dependent solution, where the theory provides a functional equation for the generating function of the time to first emptiness (Kendall (1957)), and the other with the asymptotic stationary distribution of reservoir contents, for which an explicit formula for the generating function is available (Moran (1959)).

scholarly journals Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 334 ◽  
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Fibonacci Polynomials ◽  
Recursive Sequence ◽  
Combinatorial Methods

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.

How many random digits are required until given sequences are obtained?

1982 ◽  
Vol 19 (03) ◽  
pp. 518-531 ◽  
Author(s):  
Gunnar Blom ◽  
Daniel Thorburn
Keyword(s):  
Generating Function ◽  
Waiting Time ◽  
Probability Generating Function ◽  
Recursive Formula ◽  
Fixed Number ◽  
Digit Sequence ◽  
Mean Waiting Time ◽  
The Mean ◽  
The Given

Random digits are collected one at a time until a given k -digit sequence is obtained, or, more generally, until one of several k -digit sequences is obtained. In the former case, a recursive formula is given, which determines the distribution of the waiting time until the sequence is obtained and leads to an expression for the probability generating function. In the latter case, the mean waiting time is given until one of the given sequences is obtained, or, more generally, until a fixed number of sequences have been obtained, either different sequences or not necessarily different ones. Several results are known before, but the methods of proof seem to be new.

Distribution of the Time of the First k-Record

1997 ◽  
Vol 11 (3) ◽  
pp. 273-278 ◽  
Author(s):  
Ilan Adler ◽  
Sheldon M. Ross
Keyword(s):  
Generating Function ◽  
Random Variables ◽  
Recursive Formula ◽  
Finite Set ◽  
Independent And Identically Distributed

We compute the first two moments and give a recursive formula for the generating function of the first k-record index for a sequence of independent and identically distributed random variables that take on a finite set of possible values. When the random variables have an infinite support, we bound the distribution of the index of the first k-record and show that its mean is infinite.

scholarly journals Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions

2017 ◽  
Vol 15 (1) ◽  
pp. 1156-1160 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Recursion Formula ◽  
Derivatives Of

Abstract A recursion formula for derivatives of Chebyshev polynomials is replaced by an explicit formula. Similar formulae are derived for scaled Fibonacci numbers.

scholarly journals Counting Derangements, Non Bijective Functions and the Birthday Problem

2010 ◽  
Vol 18 (4) ◽  
pp. 197-200
Author(s):  
Cezary Kaliszyk
Keyword(s):  
Explicit Formula ◽  
Recursive Formula ◽  
Finite Sets ◽  
Birthday Problem ◽  
Finite Set ◽  
One To One

Counting Derangements, Non Bijective Functions and the Birthday Problem The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].

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