Binomial Convolution Research Articles

Quotients of sequences under the binomial convolution

This paper gives expressions for the solution $\{a(n)\}$ of the equation \begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*} where $b(0)\ne 0$, that is, of the equation $a \circ b = c$ in $a$, where $\circ$ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.

A Context-Free Grammar Associated with Fibonacci and Lucas Sequences

We introduce a context-free grammar G = s ⟶ s + d , d ⟶ s to generate Fibonacci and Lucas sequences. By applying the grammar G , we give a grammatical proof of the Binet formula. Besides, we use the grammar G to provide a unified approach to prove several binomial convolutions about Fibonacci and Lucas numbers, which were given by Hoggatt, Carlitz, and Church. Meanwhile, we also obtain some new binomial convolutions.

Some notes on the algebraic structure of linear recurrent sequences

AbstractSeveral operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an R-algebra, given any commutative ring R with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these R-algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.

Asymptotics of Some Plancherel Averages Via Polynomiality Results

Consider Young diagrams of n boxes distributed according to the Plancherel measure. So those diagrams could be the output of the RSK algorithm, when applied to random permutations of the set {1,ldots ,n}. Here we are interested in asymptotics, as nrightarrow infty , of expectations of certain functions of random Young diagrams, such as the number of bumping steps of the RSK algorithm that leads to that diagram, the side length of its Durfee square, or the logarithm of its probability. We can express these functions in terms of hook lengths or contents of the boxes of the diagram, which opens the door for application of known polynomiality results for Plancherel averages. We thus obtain representations of expectations as binomial convolutions, that can be further analyzed with the help of Rice’s integral or Poisson generating functions. Among our results is a very explicit expression for the constant appearing in the almost equipartition property of the Plancherel measure.

Note on $ r $-central Lah numbers and $ r $-central Lah-Bell numbers

<abstract><p>The $ r $-Lah numbers generalize the Lah numbers to the $ r $-Stirling numbers in the same sense. The Stirling numbers and the central factorial numbers are one of the important tools in enumerative combinatorics. The $ r $-Lah number counts the number of partitions of a set with $ n+r $ elements into $ k+r $ ordered blocks such that $ r $ distinguished elements have to be in distinct ordered blocks. In this paper, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers ($ r\in \mathbb{N} $) are introduced parallel to the $ r $-extended central factorial numbers of the second kind and $ r $-extended central Bell polynomials. In addition, some identities related to these numbers including the generating functions, explicit formulas, binomial convolutions are derived. Moreover, the $ r $-central Lah numbers and the $ r $-central Lah-Bell numbers are shown to be represented by Riemann integral, respectively.</p></abstract>

Q-Binomial Convolution and Transformations of q-Appell Polynomials

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

Some result for binomial convolution sums of restricted divisor functions

Besge presented the result about the convolution sum of divisor functions. Since then Liouville obtained the generalized version of Besge's formula, which is the binomial convolution sum of divisor functions. In 2004, Hahn obtained the results about the convolution sums of ?d|n(-1)d-1d and ?d|n (-1)n=d-1d. In this paper, we present the results for the binomial convoltion sums, generalized convolution sums of Hahn, of these divisor functions.

Estimating the Use of Public Lands: Integrated Modeling of Open Populations with Convolution Likelihood Ecological Abundance Regression

We present an integrated open population model where the population dynamics are defined by a differential equation, and the related statistical model utilizes a Poisson binomial convolution likelihood. Key advantages of the proposed approach over existing open population models include the flexibility to predict related, but unobserved quantities such as total immigration or emigration over a specified time period, and more computationally efficient posterior simulation by elimination of the need to explicitly simulate latent immigration and emigration. The viability of the proposed method is shown in an in-depth analysis of outdoor recreation participation on public lands, where the surveyed populations changed rapidly and demographic population closure cannot be assumed even within a single day.

Positivity and continued fractions from the binomial transformation

AbstractGiven a sequence of polynomials$\{x_k(q)\}_{k \ges 0}$, define the transformation$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$for$n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function ofyn(q) and that ofxn(q). We also prove that the transformation preservesq-TPr+1(q-TP) property of the Hankel matrix$[x_{i+j}(q)]_{i,j \ges 0}$, in particular forr= 2,3, implying ther-q-log-convexity of the sequence$\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of typesAandB, derangement polynomials typesAandB, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strongq-log-convexity of derangement polynomials typeB, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strongq-log-convexity.

Explicit Expressions for Higher Order Binomial Convolutions of Numerical Sequences

We give explicit expressions for higher order binomial convolutions of sequences of numbers having a finite exponential generating function. Illustrations involving Cauchy, Bernoulli, and Apostol–Euler numbers are presented. In these cases, we obtain formulas easy to compute in terms of Stirling numbers.

Closed form expressions for Appell polynomials

We show that any Appell sequence can be written in closed form as a forward difference transformation of the identity. Such transformations are actually multipliers in the abelian group of the Appell polynomials endowed with the operation of binomial convolution. As a consequence, we obtain explicit expressions for higher order convolution identities referring to various kinds of Appell polynomials in terms of the Stirling numbers. Applications of the preceding results to generalized Bernoulli and Apostol–Euler polynomials of real order are discussed in detail.

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well--known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between $H_R$ equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.

Evaluating binomial convolution sums of divisor functions in terms of Euler and Bernoulli polynomials

In this paper, we provide two identities about binomial convolution sums of [Formula: see text] with [Formula: see text], which are expressed in terms of Euler and Bernoulli polynomials. A recent result of Kim, Bayad and Park turns out to be a special case of one of the two identities when [Formula: see text].

Binomial convolution and transformations of Appell polynomials

We show that the set A of Appell sequences is an abelian group under the binomial convolution. This is essentially equivalent to other approaches considered in the literature, in particular, the determinantal approach. We also define a scale transformation Tα:A→A, which is an isomorphism, and a transformation RY:A→A based on expectations with respect to a random variable Y. Using these tools, we give explicit formulas for the determinantal form of Appell sequences, as well as for the formula of representation of powers and the Srivastava–Pintér addition theorem. Various illustrative examples, mainly referring to various generalizations of Bernoulli and Euler polynomials, are discussed in detail.

Generalized Binomial Convolution of the mth Powers of the Consecutive Integers with the General Fibonacci Sequence

Abstract In this paper, we consider Gauthier’s generalized convolution and then define its binomial analogue as well as alternating binomial analogue. We formulate these convolutions and give some applications of them.

Multiplicate inverse forms of terminating hypergeometric series

The multiplicate form of Gould–Hsu's inverse series relations enables to investigate the dual relations of the Chu–Vandermonde–Gauß’s, the Pfaff–Saalschütz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identity and reciprocal relations are thus established for terminating hypergeometric series. By virtue of the duplicate inversions, we establish several dual formulae of Chu–Vandermonde–Gauß’s and Pfaff–Saalschütz's summation theorems in Section 3 and 4, respectively. Finally, the last section is devoted to deriving several identities and reciprocal relations for terminating balanced hypergeometric series from Hagen–Rothe's convolution identity in accordance with the duplicate, triplicate and multiplicate inversions.

Preserving Log-Convexity for Generalized Pascal Triangles

We establish the preserving log-convexity property for the generalized Pascal triangles. It is an extension of a result of H. Davenport and G. Pólya "On the product of two power series", who proved that the binomial convolution of two log-convex sequences is log-convex.

Concavity of entropy along binomial convolutions

Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity in $t$ of the entropy of the convolution of a probability measure $a$, which has the law of a sum of independent Bernoulli variables, by the binomial measure of parameters $n\geq 1$ and $t$.

Khinchine type inequalities with optimal constants via ultra log-concavity

We derive Khinchine type inequalities for even moments with optimal constants from the result of Walkup (J Appl Probab 13:76–85, 1976) which states that the class of log-concave sequences is closed under the binomial convolution.

Moments on Catalan numbers

By combining inverse series relations with binomial convolutions and telescoping method, moments of Catalan numbers are evaluated, which resolves a problem recently proposed by Gutiérrez et al. [J.M. Gutiérrez, M.A. Hernández, P.J. Miana, N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (1) (2008) 52–61].