Appell Sequences Research Articles

The Appell sequences of fractional type

The Appell sequences of fractional type

Symmetric Identities Involving Discrete Appell Sequences

Symmetric Identities Involving Discrete Appell Sequences

Boole-Dunkl polynomials and generalizations

Appell sequences of polynomials can be extended to the Dunkl context replacing the ordinary derivative by the Dunkl operator on the real line, and the exponential function by the Dunkl kernel. In a similar way, discrete Appell sequences can be extended to the Dunkl context; here, the role of the ordinary translation is played by the Dunkl translation, which is a much more intricate operator. Some sequences as the falling factorials or the Bernoulli polynomials of the second kind have already been extended and investigated in the mathematical literature. In this paper, we study the discrete Appell version of the Euler polynomials, usually known as Euler polynomials of the second kind or Boole polynomials. We show how to define the Dunkl extension of these polynomials (and some of their generalizations), and prove some relevant properties and relations with other polynomials and with Stirling-Dunkl numbers.

A general method to find special functions that interpolate Appell polynomials, with examples

Given an Appell sequence {Pn(x)}n=0∞ defined by means of a generating functionA(t)ext=∑n=0∞Pn(x)tnn!, we discuss a general procedure for constructing a complex function F(s,x), which is entire in s for each fixed x with Rex>0, and satisfies F(−n,x)=Pn(x) at n=0,1,2,…. The method is based on the Mellin transform and allows A(−t) to have isolated singularities on the half-line (0,∞), in contrast with other general methods that appear in the mathematical literature. We illustrate our procedure with some elucidatory examples. However, our approach cannot be used for analogously defined Appell-Dunkl sequences, a fact which has led us to include an open problem related to this case.

Several Characterizations of Δh-Doped Special Polynomials Associated with Appell Sequences

The study presented in this paper follows the line of research created by the fact that by employing the monomiality principle, new outcomes are produced. This article deals with the inducement of Δh tangent-based Appell polynomials and derivation of certain of its characterizations such as explicit form, determinant form, monomiality principle, etc. These polynomials are designed to exhibit certain symmetries themselves or to capture and describe symmetrical patterns in mathematical structures. Further, certain members of Δh Appell polynomials such as Δh Bernoulli, Euler, and Genocchi polynomials are taken, and their corresponding results are obtained.

Certain Properties and Applications of Convoluted Δh Multi-Variate Hermite and Appell Sequences

This study follows the line of research that by employing the monomiality principle, new outcomes are produced. Thus, in line with prior facts, our aim is to introduce the Δh multi-variate Hermite Appell polynomials ΔhHAm[r](q1,q2,⋯,qr;h). Further, we obtain their recurrence sort of relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families and thus operational principle satisfied by these polynomials is derived and will prove beneficial for future observations. Further, a few members of the Δh Appell polynomial family are considered and their corresponding results are derived accordingly.

Certain Properties and Applications of Δh Hybrid Special Polynomials Associated with Appell Sequences

The development of certain aspects of special polynomials in line with the monomiality principle, operational rules, and other properties and their aspects is obvious and indisputable. The study presented in this paper follows this line of research. By using the monomiality principle, new outcomes are produced, and their differential equation and series representation is obtained, which are important in several branches of mathematics and physics. Thus, in line with prior facts, our aim is to introduce the Δh hybrid special polynomials associated with Hermite polynomials denoted by ΔhHQm(u,v,w;h). Further, we obtain some well-known main properties and explicit forms satisfied by these polynomials.

SYMBOLIC APPROACH TO THE QUADRATIC DECOMPOSITION OF APPELL SEQUENCES

In this paper, we characterize the four derived sequences obtained by the symbolic approach to the quadratic decomposition of Appll sequences. Moreover, we prove that the two monic polynomial sequences associated to such quadratic decomposition are also Appell sequences.

Appell-Dunkl sequences and Hurwitz-Dunkl zeta functions

Dunkl theory on the real line involves some tools such as the Dunkl derivativeΛαf(x)=ddxf(x)+2α+12f(x)−f(−x)x or the Dunkl exponential Eα(z) that is defined in terms of the Bessel functions. Taking α=−1/2 we get Λ−1/2=d/dx and E−1/2(z)=ez, hence, the classic derivative and exponential are particular cases. In recent years, some papers have generalized, in a Dunkl sense, number theoretic concepts such as Appell sequences, and then they are called Appell-Dunkl sequences; in particular, the so called Bernoulli-Dunkl and Euler-Dunkl polynomials have been defined, among others. Here we generalize, also in a Dunkl sense, some Hurwitz or Lerch zeta functions such as ζ(s,x)=∑n=0∞1/(n+x)s and, in addition, we get properties that relate those functions, extended to the s-complex plane and evaluated at negative integers s, with Bernoulli-Dunkl and Euler-Dunkl polynomials. One of the results we get for the “Dunkl zeta function” ζα(s) isζα(1−s)=Γ(s)cos⁡(πs2)∑n=1∞1sns,Re(s)>1 (where sn are the positive zeros of the Bessel function Jα+1(x)). This equation provides a generalization of the reflection formula of the Riemann zeta function, where the function ∑n=1∞1/sns is playing a similar role as ∑n=1∞1/ns.

Recurrence relation for the Appell sequences

In this paper, we present several explicit formulas for Appell polynomials in terms of the weighted Stirling numbers of the second kind. We also provide a unified approach to obtain a three-term recurrence formula for the computation of Appell polynomials. Several examples are given to illustrate our results. As an application, we define and study a new class of polynomials that we call SPI polynomials. They are related to sums of powers of integers.

Stirling-Dunkl numbers

The Appell sequences of polynomials can be extended to the Dunkl context, where the ordinary derivative is replaced by Dunkl operator on the real line, and the exponential function is replaced by the so-called Dunkl kernel. In a similar way, the discrete Appell sequences can be extended to the Dunkl context, where the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In particular, this allows to define the falling factorial polynomials in the Dunkl context. Some numbers closely related to falling factorial are the so called Stirling numbers of the first kind and of the second kind, as well as the Bell numbers and the Bell polynomials. In this paper, we define these numbers and polynomials in the Dunkl context, and prove some of their properties.

Discrete Appell-Dunkl sequences and Bernoulli-Dunkl polynomials of the second kind

In a similar way that the Appell sequences of polynomials can be extended to the Dunkl context, where the ordinary derivative is replaced by Dunkl operator on the real line, and the exponential function is replaced by the so-called Dunkl kernel, one can expect that the discrete Appell sequences can be extended to the Dunkl context. In this extension, the role of the ordinary translation is played by the Dunkl translation, that is a much more intricate operator. In this paper, we define discrete Appell-Dunkl sequences of polynomials, and we give some properties and examples. In particular, we show which is the suitable definition for the Bernoulli polynomials of the second kind in the Dunkl context.

Advanced selection of ensemble control tools

We propose a method for generating a wide variety of increasingly complex microscopic temperature expressions in the form of functional polynomials in thermodynamic temperature. The motivation for study of such polynomials comes from thermostat theory. The connection of these polynomials with classical special functions, in particular, with Appell sequences, is revealed.

Orthogonal polynomials, biorthogonal polynomials and spline functions

A sequence of continuous real functions ( f m ) m ∈ N 0 , with exponential decay at infinity that implies their Fourier-Laplace transforms F m ( i z ) are analytic on | ℜ ( z ) | < B , z ∈ C , generates a sequence of polynomials ( Q k ) k ∈ N 0 , that are biorthogonal to the distributional derivatives μ m : = ( − 1 ) m f m ( m ) , m ∈ N 0 . The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel e x z in terms of μ ˆ m ( i z ) = z m F m ( i z ) , m ∈ N 0 . If μ m = ω P m , where P m are polynomials of degree m and ω is a weight function, Q m is a constant multiple of P m and they are orthogonal polynomials. In the case where f m are uniform or quasi-uniform B -splines, the corresponding biorthogonal polynomials exhibit properties reminiscent of Appell sequences, while in the case where f m are refinable functions they are eigenfunctions of the adjoint of the corresponding refinement operators. For nonuniform B -splines, a suitable choice of the knot sequence defines ( f m ) m ∈ N 0 in which the corresponding polynomials Q k reduce to the Jacobi polynomials and the biorthogonal spline functions μ m reduce to the corresponding weighted Jacobi polynomials, if there are no interior knots.

Properties of Partially Degenerate Complex Appell Polynomials

Degenerate versions of polynomial sequences have been recently studied to obtain useful properties such as symmetric identities by introducing degenerate exponential-type generating functions. As part of our continued work in degenerate versions of generating functions, we subsequently present our study on degenerate complex Appell polynomials by considering a partially degenerate version of the generating functions of ordinary complex Appell polynomials in this paper. We only consider partially degenerate generating functions to retain the crucial properties of the Appell sequence, and we present useful identities and general properties by splitting complex values into their real and imaginary parts; moreover, we provide several explicit examples. Additionally, the differential equations satisfied by degenerate complex Bernoulli and Euler polynomials are derived by the quasi-monomiality principle using Appell-type polynomials.

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 Note on the Degenerate Type of Complex Appell Polynomials

In this paper, complex Appell polynomials and their degenerate-type polynomials are considered as an extension of real-valued polynomials. By treating the real value part and imaginary part separately, we obtained useful identities and general properties by convolution of sequences. To justify the obtained results, we show several examples based on famous Appell sequences such as Euler polynomials and Bernoulli polynomials. Further, we show that the degenerate types of the complex Appell polynomials are represented in terms of the Stirling numbers of the first kind.

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.

Fractional calculus and generalized forms of special polynomials associated with Appell sequences

Abstract In this article, an operational definition, generating function, explicit summation formula, determinant definition and recurrence relations of the generalized families of Hermite–Appell polynomials are derived by using integral transforms and some known operational rules. An analogous study of these results is also carried out for the generalized forms of the Hermite–Bernoulli and Hermite–Euler polynomials.

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.