Polynomial Sequences Of Binomial Type Research Articles

An infinite dimensional umbral calculus

The aim of this paper is to develop foundations of umbral calculus on the space D′ of distributions on Rd, which leads to a general theory of Sheffer polynomial sequences on D′. We define a sequence of monic polynomials on D′, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on D′ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on R of binomial type to a polynomial sequence of binomial type on D′, and a lifting of a Sheffer sequence on R to a Sheffer sequence on D′. Examples of lifted polynomial sequences include the falling and rising factorials on D′, Abel, Hermite, Charlier, and Laguerre polynomials on D′. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.

A New Recurrence Relation and Related Determinantal form for Binomial Type Polynomial Sequences

For polynomial sequence of binomial type we propose a new recurrence formula equivalent to a determinantal form of the sequence, from which known and unknown properties are deducted. Relationships whit $${\delta}$$ –functionals, $${\delta}$$ –operators and generating function are pointed out. Finally, some classical examples are studied, highlighting interesting determinantal identities.

Polynomial Sequences of Binomial-Type Arising in Graph Theory

In this paper, we show that the solution to a large class of "tiling'' problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, and that certain sets of these polynomials satisfy binomial-type recurrences. We exhibit generalizations of this theorem to higher dimensions and other lattices. Finally, we apply the techniques developed in this paper to resolve an open question about the structure of coefficients of chromatic polynomials of certain grid graphs (namely that they also satisfy a binomial-type recurrence).

An algebraic approach to Sheffer polynomial sequences

A matrix approach to Sheffer polynomial sequences is proposed; in particular, two different determinantal forms of Sheffer sequences are given, the one as the function of a polynomial sequence of binomial type and the other as the function of the canonical base xi. The equivalence with the classical definitions of Sheffer and Roman and Rota is proven. Then, elementary matrix algebra tools are employed to reveal the known and unknown properties of Sheffer polynomials. Finally, classical and non-classical examples are also considered.

Matrix Representations of a Special Polynomial Sequence in Arbitrary Dimension

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows for their recursive construction in the same way as for complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.

Schur type functions associated with polynomial sequences of binomial type

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover, this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras.

Set maps, umbral calculus, and the chromatic polynomial

Some important properties of the chromatic polynomial also hold for any polynomial set map satisfying p S ( x + y ) = ∑ T ⊎ U = S p T ( x ) p U ( y ) . Using umbral calculus, we give a formula for the expansion of such a set map in terms of any polynomial sequence of binomial type. This leads to some new expansions of the chromatic polynomial. We also describe a set map generalization of Abel polynomials.

Endomorphisms of the binomial coalgebra

The algebra of formal Hurwitz series has been recently investigated in view to potential applications to differential algebra. In the case where the ground ring is a reduced ring of prime characteristic, this paper describes continuous endomorphisms of the algebra of Hurwitz series—or equivalently endomorphisms of the binomial coalgebra. This is done by investigating the coradical filtration of this coalgebra and components of linear maps between some graded modules that naturally arise in the problem. Our problem is tantamount to that of determining all polynomial sequences of binomial type. As such, it contains the study of sequences built by Carlitz in the context of his discovery of the Carlitz module.

Generalized Rook Polynomials

Generalizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given.

Multivariate shift invariant operators

An unexpected connection between a certain class of exponential approximation operators and polynomial sequences of binomial type was discovered by Ismail ( J. Approx. Theory (3) 23 (1978), 117–186). In this paper we present a multivariate analogue of shift invariant operators connected with Sheffer type polynomials.

Operatorenkalk�l �ber 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.)

Expansion formulas. II. Variations on a theme

In “Expansion Formulas, I” [S. A. Joni, J. Math. Anal. Appl. 81 (1981)], it was shown that the Steffensen formula or polynomial sequences of binomial type gives rise to a method for generating a certain class of expansion identities. Special cases of this class of identities were studied by Carlitz [ SIAM J. Appl. Math. 26 (1974), 431–436; 8 (1977), 320–336]. Since the Umbral calculus for polynomial sequences of binomial type has been generalized to encompass the theories of composition sequences [A. M. Garsia and S. A. Joni, Comm. Algebra, in press] and factor sequences [S. Roman and G.-C. Rota, Adv. in Math. 27 (1978), 95–188], we herein extend the results of part I to these two more general settings.

Sur l'Inversion et l'Iteration Continue des Séries Formelles

This paper deals with the composition of normalised formal power series, in one variable, over an arbitrary field K of characteristic zero. A suitable group structure B ⊙ on the set B of polynomial sequences of binomial type is introduced. This group is used first to obtain many formal variants of the classical Lagrange inversion formula (without using any complex integration). Secondly, via one-parameter subgroups of B ⊙ , iteration (i.e., successive composition) of normalised formal power series is studied in detail for arbitrary orders s ∈ K (“continuous” iteration). The case s = −1 coincides with power series inversion. Many new formulas are derived in the course of the text. The end of the work contains suggestions for future research.

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.

A short proof of the Lagrange-Good formula

A new proof of Good's generalization to several variables of the Lagrange inversion formula is given, which is mainly based on methods used in Rota's theory of polynomial sequences of binomial type.

The multi-indexed partitional

In this paper we present the multi-indexed partitional. The structure is a generalization of the partitional of Foata and Schützenberger, and gives a combinatorial interpretation for certain multivariate polynomial sequences of binomial type.

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.

Rook Theory. II: Boards of Binomial Type

We explore the relation between rook theory of Ferrers boards and polynomial sequences of binomial type. Recursion and explicit formulas for the rook numbers of Riordan’s trapezoidal boards are derived. It is shown that some classic sequences of rook and factorial polynomials are of binomial type. A class of boards corresponding to Abel’s theorem and its generalizations are constructed. The results are developed from both algebraic and combinatorial viewpoints.