Interpolation and Combinatorial Functions

Using some basic results about polynomial interpolation, divided differences, and Newton polynomial sequences we develop a theory of generalized binomial coefficients that permits the unified study of the usual binomial coefficients, the Stirling numbers of the second kind, the q-Gaussian coefficients, and other combinatorial functions. We obtain a large number of combinatorial identities as special cases of general formulas. For example, Leibniz's rule for divided differences becomes a Chu-Vandermonde convolution formula for each particular family of generalized binomial coefficients.     

The Combinatorics of Polynomial Sequences

We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of U-graph and give examples of binomial prefabs of U-graphs. The umbral composition of U-graphs provides an interpretation of umbral composition of polynomial sequences in Sees. 3 and 5. Rota's interpretation of the Stirling numbers of the first kind as sums of the Mobius function in the partition lattice inspired our model for inverse sequences of binomial type in Sec. 4. Section 6 contains combinatorial proofs of several operator-theoretic results. The actions of shift operators and delta operators are explained in set-theoretic terms. Finally, in Sec. 6 we give a model for cross sequences and Sheffer sequences which is consistent with their decomposition into sequences of binomial type. This provides an interpretation of shift-invariant operators. Of course, all of these interpretations require that the coefficients involved be integer and usually non-negative as well.     

Sequences of polynomials of fractional binomial type

Sequences of polynomials which satisfy a binomial theorem involving fractional binomial coefficients can be characterized as umbral left inverses of singular sequences of binomial type.     

Quasi-binomial coefficients stemming from Nakayama algebras

It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.     

Generalized Levinson-Durbin sequences, binomial coefficients and autoregressive estimation

For a discrete time second-order stationary process, the Levinson-Durbin recursion is used to determine the coefficients of the best linear predictor of the observation at time k+1, given k previous observations, best in the sense of minimizing the mean square error. The coefficients determined by the recursion define a Levinson-Durbin sequence. We also define a generalized Levinson-Durbin sequence and note that binomial coefficients form a special case of a generalized Levinson-Durbin sequence. All generalized Levinson-Durbin sequences are shown to obey summation formulas which generalize formulas satisfied by binomial coefficients. Levinson-Durbin sequences arise in the construction of several autoregressive model coefficient estimators. The least squares autoregressive estimator does not give rise to a Levinson-Durbin sequence, but least squares fixed point processes, which yield least squares estimates of the coefficients unbiased to order 1/T, where T is the sample length, can be combined to construct a Levinson-Durbin sequence. By contrast, analogous fixed point processes arising from the Yule-Walker estimator do not combine to construct a Levinson-Durbin sequence, although the Yule-Walker estimator itself does determine a Levinson-Durbin sequence. The least squares and Yule-Walker fixed point processes are further studied when the mean of the process is a polynomial time trend that is estimated by least squares.     

Operatorenkalkül über freien Monoiden I: Strukturen

In part I algebraic structures (esp. rings) on the sets of polynomials and formal power series on an at most countable alphabetA are considered. Given a partial order onA the words ofA ^* are mixed together in consistence with it. It is shown that the structures derived are associative iff the given partial order is of linear type. The coefficients appearing at these operations are identified as generalizations of the ordinary binomial coefficients and a number of relations involving them are listed up.(Part II will bring a generalization ofRota's theory of polynomial sequences of binomial type to the structures studied in I.In Part III the theory of special binomial systems will be continued until the analogue of Lagrange inversion and a short development of generalized Sheffer polynomials will be given).     

On a kind of curious binomial identity

As a generalization of Calkin's identity and its alternating form, we compute a kind of binomial identity involving some real number sequences and a partial sum of the binomial coefficients, from which many interesting identities follow.     

Ballots and plane trees

The correspondence of certain plane trees and binary sequences reported by D. A. Klarner in [1], and a ballot interpretation of the latter, are used to make an independent evaluation of the number of classes of isomorphic, (k + 1)-valent, planted plane trees with kn + 2 points. This provides an interesting multivariable identity for binomial coefficients.     

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers, are Hausdorff moment sequences. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.     

M-bonomial coefficients and their identities

In this note, we introduce M-bonomial coefficients or (M-bonacci binomial coefficients). These are similar to the binomial and the Fibonomial (or Fibonacci–binomial) coefficients and can be displayed in a triangle similar to Pascal's triangle from which some identities become obvious.     

Some properties of a class of sparse polynomials

We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.     

Pairs of Lattice Paths and Positive Trigonometric Sums

Ismail et al. (Constr. Approx. 15:69–81, 1999) proved the positivity of some trigonometric polynomials with single binomial coefficients. In this paper, we prove some similar results by replacing the binomial coefficients with products of two binomial coefficients.     

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.      

Asymptotic expansions of the binomial coefficients

A general formulae for the asymptotic expansion of not centered binomial coefficients are derived and some useful estimates of the binomial coefficients are presented. The sum of the binomial coefficients is also studied.     

On the properties of Lucas numbers with binomial coefficients

In this study, some new properties of Lucas numbers with binomial coefficients have been obtained to write Lucas sequences in a new direct way. In addition, some important consequences of these results related to the Fibonacci numbers have been given.     

On the exponent average of integer sequences

Let ω(m) denote the number of distinct prime factors of the integerm, let Ω(m) be the number of prime factors ofm counted with multiplicities. The exponent averageA(m) is defined byA(m)=Ω(m)/ω(m) form>1, andA(1)=1. If (m _n) is a sequence of positive integers, we can study the asymptotic exponent average lim _n→∞ A(m_n) (if it exists) resp. lim sup and lim inf. In this article, we consider exponent averages for general sequences, and particularly for sequences of binomial coefficients as well as the divisor function. One of the many results on binomial coefficients is that $$\mathop {\lim }\limits_{n \to \infty } A\left( {\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right)} \right) = 1,$$ which shows that these binomial coefficients are almost squarefree. For the divisor functiond(n), we prove for instance $$\mathop {\lim \sup }\limits_{n \to \infty } \frac{{A(d(n))\log \log n}}{{\log n}} = 1.$$      

Combinatorics of Nilpotents in Symmetric Inverse Semigroups

We show how several famous combinatorial sequences appear in the context of nilpotent elements of the full symmetric inverse semigroup . These sequences appear either as cardinalities of certain nilpotent subsemigroups or as the numbers of special nilpotent elements and include the Lah numbers, the Bell numbers, the Stirling numbers of the second kind, the binomial coefficients and the Catalan numbers.AMS Subject Classification: 05A15, 20M18, 20M20, 05A19.     

A construction of low-discrepancy sequences involving finite-row digital (t,s)-sequences

This paper introduces a construction principle for generating matrices of digital sequences over a finite field $\mathbb{F }_q$ , which is based on sequences of polynomials and their representations in terms of powers of nonconstant polynomials. For the most basic polynomial sequence, $(x^r)_{r\ge 0}$ , the representations in terms of powers of linear polynomials yield, within this construction principle, the Pascal matrices, which consist of binomial coefficients and were earlier introduced by Faure for finite prime fields and by Niederreiter for finite field extensions. Generally, for binomial type sequences of polynomials an interesting relation between the generating matrices is worked out, and further examples of generating matrices are given, which contain combinatorial magnitudes as, e.g., binomial coefficients, Stirling numbers of the first kind, Stirling numbers of the second kind, and Bell numbers. Moreover, within this construction principle, explicit constructions of finite-row generating matrices of digital $(t,s)$ -sequences are presented, which were so far only known for $t$ equal to $0$ .     

On the Properties of Iterated Binomial Transforms for the Padovan and Perrin Matrix Sequences

In this study, we apply “r” times the binomial transform to the Padovan and Perrin matrix sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we give the relationships of between iterated binomial transforms for Padovan and Perrin matrix sequences.     

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.