关于Lucas数列同余性质的研究

将二项式系数的性质应用到Lucas数列的研究中,并结合Fibonacci数列与Lucas数列的恒等式得到几个有趣的Lucas数列的同余式.     

Sequences of functions of binomial type

The theory of binomial enumeration leads to sequences of functions of binomial type which are not polynomials. The results of Mullin-Rota for these sequences are developed and a ring structure on the set of sequences is studied.     

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.     

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 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.     

二项式型自反多项式

In this paper, we define the self-inverse sequences related to sequences of polynomials of binomial type, and give some interesting results of these sequences. Moreover, we study the self-inverse sequences related to the Laguerre polynomials.     

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.     

Applications of a binoomial-type identity

A generalization of the binomial theorem is proved for sequences of polynomials of binomial type. The equivalence of two formulas which count group-invariant partitions is shown to be a consequence of this theorem. The theorem is also used to demonstrate the equivalence of two formulas for rook polynomials.     

Some comments on Rota's Umbral Calculus

Rota's Umbral Calculus is put in the context of general Fourier analysis. Also, some shortcuts in the proofs are illustrated and a new characterization of sequences of binomial type is given. Finally it is shown that there are few (classical) orthogonal polynomials of binomial type.     

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.     

序列的一种无组合系数的根表示

本文在讨论了组合数C_x~k等在GF(q)上的多元多项式表示的基础上,给出了序列的一种避免组合系数的根表示法,并利用它对两个有重根的反馈多项式生成序列之积的线性复杂性进行了讨论.     

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.

     

Frobenius Endomorphisms in the Umbral Calculus

Frobenius operators F_n are introduced on sequences of binomial type. The Laguerre polynomials are essentially characterized by the property that F_n coincides with n-fold binomial convolution.     

Operatorenkalkül über freien Monoiden III: Lagrangeinversion und Sheffersysteme

In this part the relations between the theory of binomial systems for noncommuting variables (of part II) and the results ofRota, Cigler et al. for the commutative case are studied in some detail. Afterwards for binomial systems of diagonal type there are given generalizations of the Rodrigues formula and the theorem ofLagrange-Good. A short development of Sheffer sequences follows. Finally the results are extended to binomial systems the structure of which is determined by any partial order of linear type.     

Operatorenkalkül über freien Monoiden II: Binomialsysteme

In this part starting from a generalization of the binomial theorem a development of Rota's theory of polynomial sequences of binomial type to the case of countably many noncommuting variables is given. Translation invariance of operators gives the relation to the formal power series considered inI. For a special class of binomial systems there are given a number of characterizations, such as a generalized Rodrigues formula. In the case of an analogue of the Newton polynomials those are used for a study of generalized Stirling numbers. (In III the theory of binomial systems of diagonal type will be continued until an analogue of Lagrange inversion and a short development of the theory of generalized Sheffer polynomials will be given.)     

CHARACTERIZATION FOR BINOMIAL SEQUENCES AMONG RENEWAL SEQUENCES

In this paper a binomial sequence is defined as a special sequence whose renewal lifes are identically distributed with a common geometric distribution. Therefore, it can be regarded as the discrete version of a Poisson process. Mainly, we discuss the characterization problem associated with binomial sequences. First, we sketch the properties of some important quantities of a renewal sequence. The emphasis of discussion is laid on the current life, the residual life and the total life. Then, we describe three main approaches to identify a geometric distribution. Finally, based on these concepts and techniques, we give a series of characterization theorems for a binomial sequence. These results are quite similar to those obtained for a Poisson process.     

n-Colour self-inverse compositions

MacMahon’s definition of self-inverse composition is extended ton-colour self-inverse composition. This introduces four new sequences which satisfy the same recurrence relation with different initial conditions like the famous Fibonacci and Lucas sequences. For these new sequences explicit formulas, recurrence relations, generating functions and a summation formula are obtained. Two new binomial identities with combinatorial meaning are also given.     

Multivariate Exponential Operators

An unexpected connection between a certain class of exponential approximation operators and polynomial sequences of binomial type was discovered by Ismail. Building on this result, we present a multivariate analogue of these exponential operators.     

Some identities involving the binomial sequences

Using the exponential generating function and the Bell polynomials, we obtain several new identities for the binomial sequences. As applications, some interesting identities are established for the Abel polynomials, exponential polynomials and factorial powers.