Appell Polynomials Research Articles

Appell polynomials and their relatives III. Conditionally free theory

We extend to the multivariate noncommutative context the descriptions of a “once-stripped” probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map $Φ$ on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bozejko, Leinert, and Speicher. This theory includes the free and Boolean probability theories as extreme cases. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map $Φ$ above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath–Segall polynomials, their combinatorics, and Hilbert space representations.

On Some Characteristics of the Multivariate Analogues of the Chebyshev–Hermite Polynomials

This paper investigates some properties of the multivariate analogues of the Chebyshev–Hermite polynomials and proves that they are the Appel polynomials (under a respective generalization of this notion). Recurrent formulae for multivariate analogues of the Chebyshev–Hermite polynomials are also obtained.

Hermite-based Appell polynomials: Properties and applications

By employing certain operational methods, the authors introduce Hermite-based Appell polynomials. Some properties of Hermite–Appell polynomials are considered, which proved to be useful for the derivation of identities involving these polynomials. The possibility of extending this technique to introduce Hermite-based Sheffer polynomials (for example, Hermite–Laguerre and Hermite–Sister Celine's polynomials) is also investigated.

On Appell Sets and the Fueter-Sce Mapping

It is proved, that the recently discussed Appell polynomials in Clifford algebras are the Fueter-Sce extension of the complex monomials z^k. Furthermore, it is shown, for which complex functions the Fueter-Sce extension and the extension method using Appell polynomials coincide.

Determinant representations of Appell polynomial sequences

In this paper, we develop a new representation for Appell polynomial sequence via determinants. In addition, we demonstrate a direct application of this determinant formula (representation) for Appell sequence by deriving a determinant formula for a power series of a function f (t) in terms of the power series of its algebraic inverse function 1/f (t) . Mathematics subject classification (2000): 15A15, 05A40, 11B68.

Appell polynomials and their relatives II. Boolean theory

The Appell-type polynomial family corresponding . to the simplest non-commutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal non-commutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolvent-type form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the Belinschi-Nica and Bercovici-Pata maps, conditional freeness, and the Laha-Lukacs type characterization. A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi parameters under convolution, the relationship between the Jacobi parameters and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

Quadratic decomposition of Appell sequences

We proceed to the quadratic decomposition of Appell sequences and we characterise the four derived sequences obtained by this approach. We prove that the two monic polynomial sequences associated to such quadratic decomposition are also Appell sequences with respect to another (lowering) operator, which we call as F ɛ , where either ɛ = 1 or - 1 . Thus, we introduce and develop the concept of the Appell polynomial sequences with respect to the operator F ɛ (where, ɛ is a parameter belonging to the field of the complex numbers): the F ɛ -Appell sequences. The orthogonal polynomial sequences that are also F ɛ -Appell correspond to the Laguerre sequences with parameter ɛ / 2 . Indeed, this brings an entirely new characterisation of the Laguerre sequences.

Formal groups, Bernoulli-type polynomials and L-series

A new construction relating formal groups, a class of Appell polynomials called the universal Bernoulli polynomials and a family of Dirichlet L-series is proposed. Universal Bernoulli χ-numbers as well as generalized Riemann–Hurwitz zeta functions are introduced. To cite this article: P. Tempesta, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

Spectral order and isotonic differential operators of Laguerre–Pólya type

The spectral order on R n induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Polya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space l∞ of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Polya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

Turan's inequality for appell polynomials

We give some necessary and sufficient conditions for the class of Appell polynomials to satisfy well-known Turan's inequality. Among the other corollaries, we apply our results to some classes of orthogonal polynomials.

CHARACTERIZATION OF OPERATORS ON A KONDRATIEV SPACE IN THE NON-GAUSSIAN SETTING: BIORTHOGONAL APPROACH

A linear bounded operator on a test space of white noise functionals can be characterized in terms of growth conditions imposed on the operator's symbol. This paper extends such characterization to operators on the Kondratiev's test space [Formula: see text] of random variables over a general non-Gaussian probability space. We follow the biorthogonal approach of Yu. Daletsky and his colleagues. In particular, the test space under consideration is generated by the system of multivariate Appell polynomials defined on the underlying infinite dimensional co-nuclear probability space. We further extend this biorthogonal approach to operators by providing a biorthogonal chaos decomposition for operators and by giving a biorthogonal construction for an operator's symbol.

On an Effective Solution of the Optimal Stopping Problem for Random Walks

We find a solution of the optimal stopping problem for the case when a reward function is an integer power function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval $\{0,1,\ldots,T\}$ converges with an exponential rate as $T\to\infty$ to the limit under the assumption that jumps of the random walk are exponentially bounded.

Another look at the Picard-Lefèvre formula for finite-time ruin probabilities

In the compound Poisson risk model, with discrete claim size distribution, Picard and Lefèvre [Picard, P., Lefèvre, C., 1997. The probability of ruin in finite-time with discrete claim size distribution. Scand. Actuarial J. 1, 58–69] derived a formula to compute the finite-horizon ruin probability. Here, some alternatives to this formula are proposed: exact recursive formulas which provide the distribution of time to ruin at once and a Seal-type formula which only involve probabilistic quantities. Depending on the comparison between the initial reserve and the total premium up to the finite horizon, their different interests are discussed by comparing their performances. The numerical stability of the formulas is then investigated, and disagreements in the existing literature about the detection of critical values are explained. Formal convolutions for pseudo-compound distributions are introduced, and a theorem is stated in order to switch between formulas based on Appell polynomials and Seal-type formulas. This also provides a derivation of the Picard–Lefèvre formula from sample path properties.

MULTIDIMENSIONAL EXTENSIONS OF THE BERNOULLI AND APPELL POLYNOMIALS

Multidimensional extensions of the Bernoulli and Appell polynomials are defined generalizing the corresponding generating functions, and using the Hermite-Kamp´e de F´eriet (or Gould-Hopper) polynomials. Furthermore the differential equations satisfied by the corresponding 2D polynomials are derived exploiting the factorization method, introduced in [15].

Laguerre-type exponentials and generalized Appell polynomials

General classes of two variables Appell polynomials are introduced by exploiting properties of an iterated isomorphism, related to the so-called Laguerre-type exponentials. Further extensions to the multi-index and multivariable cases are mentioned.

A finite-time ruin probability formula for continuous claim severities

An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.

A finite-time ruin probability formula for continuous claim severities

An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.

MULTIDIMENSIONAL EXTENSIONS OF THE BERNOULLI AND APPELL POLYNOMIALS

Multidimensional extensions of the Bernoulli and Appell polynomials are defined generalizing the corresponding generating functions, and using the Hermite-Kamp´e de F´eriet (or Gould-Hopper) polynomials. Furthermore the differential equations satisfied by the corresponding 2D polynomials are derived exploiting the factorization method, introduced in [15].

Generalizations of the Bernoulli and Appell polynomials

We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag‐Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite‐Kampé de Fériet (or Gould‐Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.

Appell polynomials and logarithmic convexity

In this article we give necessary and sufficient conditions for logarithmic convexity of some sequences of Appell polynomials. Then we apply our results to Turan’s type polynomial inequalities. Precise upper and lower bounds for this class of polynomials are also determined and asymptotic behavior of

as well.