Sheffer Polynomials Research Articles

Convergence Rate of Szász Operators Involving Boas–Buck-Type Polynomials

This paper considers a linking of the well-known Szasz operators with the orthogonal polynomials, e.g., Boas–Buck-type polynomials. These polynomials include several important cases such as Brenke-type polynomials, Sheffer polynomials and Appell polynomials. We estimate the approximation degree of these operators for bounded variation functions by applying some methods and techniques of the theory of probability. Further, we extend our study to include the Voronovskaya and Gruss–Voronovskaya-type theorems in the quantitative form by utilizing the order of convergence of the sixth order moment.

ADJOINTNESS FOR SHEFFER POLYNOMIALS

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them have been studied. In this article new sets of Sheffer polynomials are introduced defining a sort of adjointness property. As an application, we show the adjoint set of Actuarial polynomials and derive their main characteristics.

Finding Determinant Forms of Certain Hybrid Sheffer Sequences

In this article, the integral transform is used to introduce a new family of extended hybrid Sheffer sequences via generating functions and operational rules. The determinant forms and other properties of these sequences are established using a matrix approach. The correspondingresults for the extended hybrid Appell sequences are also obtained. Certain examples in terms of the members of the extended hybrid Sheffer and Appell sequences are framed. By employing operational rules, the identities involving the Lah, Stirling and Pascal matrices are derived for the aforementioned sequences.

A new approach to Legendre-truncated-exponential-based Sheffer sequences via Riordan arrays⋆

The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncated-exponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations.

A Class of Sheffer Sequences of Some Complex Polynomials and Their Degenerate Types

We study some properties of Sheffer sequences for some special polynomials with complex Changhee and Daehee polynomials introducing their complex versions of the polynomials and splitting them into real and imaginary parts using trigonometric polynomial sequences. Moreover, considering their degenerate types of Sheffer sequences based on umbral composition, we present some useful expressions, properties, and examples about complex versions of the degenerate polynomials.

Operational rules and generalized special polynomials

Abstract In this work, the generalized family of Hermite–Sheffer polynomials is introduced by using Euler’s integral and operational rules. Furthermore, some properties are established. In particular, generating function and determinant definition for the generalized Hermite–Sheffer polynomials are obtained. Some examples are also considered and their corresponding results are established.

Finding hybrid families of special matrix polynomials associated with Sheffer sequences

In this paper, three index three variable Hermite matrix based Sheffer polynomials (3I3VHMSP) are introduced by algebraic decomposition of exponential operators. The operational methods combined with the monomiality principle can be used to introduce 3I3VHMSP and also to establish rules of operational nature, framing the special polynomials within the context of particular solutions of generalized forms of partial differential equations of evolution type. The Appell and Sheffer sequences along with the operational formalism offer a powerful tool for investigation of the properties of a wide class of polynomials. Further, operational representation providing connections between 3I3VHMSP families and the known special polynomials are established, which are used to derive new identities and the results for the members of these new families. The approach presented is general.

Riordan arrays and related polynomial sequences

In this paper, we generalize the results due to Luzón and Morón, and present a characterization of the generalized Riordan arrays. Using this characterization, we establish a recurrence for the generalized Sheffer sequences, and study some special types of polynomial sequences, including the generalized Lucas u-sequence and v-sequence, and the generalized Humbert polynomial sequence. In particular, by appealing to the factorizations of Riordan arrays, the umbral composition of polynomial sequences and the Hadamard product of series, we establish various relations among these polynomial sequences. Thus, we give a unified approach to some well-known polynomial sequences appeared in combinatorics and the theory of special functions.

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

For certain Sheffer sequences (sn)n=0∞ on C, Grabiner (1988) proved that, for each α∈[0,1], the corresponding Sheffer operator zn↦sn(z) extends to a linear self-homeomorphism of Eminα(C), the Fréchet topological space of entire functions of order at most α and minimal type (when the order is equal to α>0). In particular, every function f∈Eminα(C) admits a unique decomposition f(z)=∑n=0∞cnsn(z), and the series converges in the topology of Eminα(C). Within the context of a complex nuclear space Φ and its dual space Φ′, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on Φ′. In particular, for Φ=Φ′=Cn with n≥2, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space Φ′, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of Eminα(Φ′) when α>1. The latter result is new even in the one-dimensional case.

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown.

A note on Appell sequences, Mellin transforms and Fourier series

A large class of Appell polynomial sequences {pn(x)}n=0∞ are special values at the negative integers of an entire function F(s,x), given by the Mellin transform of the generating function for the sequence. For the Bernoulli and Apostol-Bernoulli polynomials, these are basically the Hurwitz zeta function and the Lerch transcendent. Each of these have well-known Fourier series which are proved in the literature using various techniques. Here we find the latter Fourier series by directly calculating the coefficients in a straightforward manner. We then show that, within the context of Appell sequences, these are the only cases for which the polynomials have uniformly convergent Fourier series. In the more general context of Sheffer sequences, we find that there are other polynomials with uniformly convergent Fourier series. Finally, applying the same ideas to the Fourier transform, considered as the continuous analog of the Fourier series, the Hermite polynomials play a role analogous to that of the Bernoulli polynomials.

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.

Product of Sheffer sequences: properties and examples

This article is written with an objective to explore the product of two Sheffer sequences. This article is an attempt to explore such type of product which extends the possibility to consider hybrid type Sheffer polynomials. It is important to remark that although this product can be viewed as the umbral composition of two Sheffer sequences but the results related to this product cannot be deduced from the results of the Sheffer sequences. The set of this product of Sheffer sequences is also a non-abelian group and thus convoluting different members of Sheffer class allows us to consider a number of hybrid type special polynomials as its members. Certain results for this class including the quasi-monomiality and determinant form are established. The article concludes with the possibility of considering the product of $n$-Sheffer sequences.

Approximation by Chlodowsky variant of Szász operators involving Sheffer polynomials

Approximation by Chlodowsky variant of Szász operators involving Sheffer polynomials

On Sheffer polynomial families

Attention is focused to particular families of Sheffer polynomials which are different from the classical ones because they satisfy non-standard differential equations, including some of fractional type. In particular Sheffer polynomial families are considered whose characteristic elements are based on powers or exponential functions.

Approximation by generalized integral Favard-Szász type operators involving Sheffer polynomials

This article deals with the approximation properties of a generalization of an integral type operator in the sense of Favard-Sz?sz type operators including Sheffer polynomials with graphics plotted using Maple. We investigate the order of convergence, in terms of the first and the second order modulus of continuity, Peetre?s K-functional and give theorems on convergence in weighted spaces of functions by means of weighted Korovkin type theorem. At the end of the work, we give some numerical examples.

Combinatorial identities involving the central coefficients of a Sheffer matrix

Given m ? N, m ? 1, and a Sheffer matrix S = [sn,k]n,k?0, we obtain the exponential generating series for the coefficients (a+(m+1)n a+mn)-1 sa+(m+1)n,a+mn. Then, by using this series, we obtain two general combinatorial identities, and their specialization to r-Stirling, r-Lah and r-idempotent numbers. In particular, using this approach, we recover two well known binomial identities, namely Gould's identity and Hagen-Rothe's identity. Moreover, we generalize these results obtaining an exchange identity for a cross sequence (or for two Sheffer sequences) and an Abel-like identity for a cross sequence (or for an s-Appell sequence). We also obtain some new Sheffer matrices.

Some results on hybrid relatives of the Sheffer polynomials via operational rules

Some results on hybrid relatives of the Sheffer polynomials via operational rules

Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind

In this paper, we derive five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices. Then we apply twelve basic identities for Sheffer polynomials, seven from previous results, to degenerate Bernoulli polynomials and Korobov polynomials of the first kind and get some new identities. In addition, letting $\lambda~\rightarrow~0$ in such identities gives usthose for Bernoulli polynomials and Bernoulli polynomials of the second kind.

New Bell–Sheffer Polynomial Sets

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear.