Symmetric Polynomial Sequences Research Articles

On the Chebyshev Polynomials and Some of Their Reciprocal Sums

In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann ζ-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

We characterize all the multiple orthogonal three-fold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as [Formula: see text]-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a [Formula: see text]-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them [Formula: see text]-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

Regular Sequences of Symmetric Polynomials

A set of n homogeneous polynomials in n variables is a regular sequence if the associated polynomial system has only the obvious solution (0,0,...,0). Denote by p\_\_k(n) the power sum symmetric polynomial in n variables x\_1\_k+x\_2\_k+...+ x\_\_n\_\_k. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets A ⊂ N\* of cardinality n such that the set of polynomials p\_\_a(n) with a ∈ A is a regular sequence. We prove that a necessary condition is that n! divides the product of the degrees of the elements of A. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already for n = 3. Given positive integers a < b < c with gcd (a,b,c) = 1, we conjecture that p\_\_a(3), p\_\_b(3), p\_\_c(3) is a regular sequence if and only if abc ≡ 0 (mod 6). We provide evidence for the conjecture by proving it in several special instances.

A basic class of symmetric orthogonal functions using the extended Sturm–Liouville theorem for symmetric functions

By using the extended Sturm–Liouville theorem for symmetric functions, we introduced a basic class of symmetric orthogonal polynomials (BCSOP) in a previous paper. The mentioned class satisfies a differential equation of the form x 2 ( px 2 + q ) Φ n ″ ( x ) + x ( rx 2 + s ) Φ n ′ ( x ) - ( n ( r + ( n - 1 ) p ) x 2 + ( 1 - ( - 1 ) n ) s / 2 ) Φ n ( x ) = 0 and contains four main sequences of symmetric orthogonal polynomials. In this paper, again by using the mentioned theorem, we introduce a basic class of symmetric orthogonal functions (BCSOF) as a generalization of BCSOP and obtain its standard properties. We show that the latter class satisfies the equation x 2 ( px 2 + q ) Φ n ″ ( x ) + x ( rx 2 + s ) Φ n ′ ( x ) - ( α n x 2 + ( 1 - ( - 1 ) n ) β / 2 ) Φ n ( x ) = 0 , in which β is a free parameter and - α n denotes eigenvalues corresponding to BCSOF. We then consider four sub-classes of defined orthogonal functions class and study their properties in detail. Since BCSOF is a generalization of BCSOP for β = s , the four mentioned sub-classes respectively generalize the generalized ultraspherical polynomials, generalized Hermite polynomials and two other finite sequences of symmetric polynomials, which were introduced in the previous work.

A basic class of symmetric orthogonal polynomials using the extended Sturm–Liouville theorem for symmetric functions

In this research, by applying the extended Sturm–Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on ( − ∞ , ∞ ) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.

Sequences of Symmetric Polynomials and Combinatorial Properties of Tableaux

In 1977 G. P. Thomas showed that the sequence of Schur polynomials associated to a partitionλcan be comfortably generated from the sequence of variablesx=(x1, x2, x3, …) by the application of mixed Baxter/multiplication operators, which in turn can be easily computed from the set SYT(λ) of standard Young tableaux of shapeλ. We generalize this construction, thereby making possible the explicit and effective computation of the Hall–Littlewood, Jack, and Macdonald polynomials used in representation theory, combinatorics, multivariate statistics, and quantum algebra. These generalized formulas have a pleasing recursive structure with respect to the Young lattice and they can easily be specialized to yield “skew” forms in all cases and “super” forms in the Schur case. We introduce and investigate: (1) the “descent polynomial of a partitionλ,” which arises naturally in the enumeration of semistandard Young tableaux of shapeλ; (2) the Boolean latticeG(ζ) associated to anyζ∈SYT(λ), which is fundamental for the “weighted” generalization of Thomas' approach to Schur polynomials; and (3) an action of the symmetric groups on semistandard Young tableaux, which is connected with Knuth's combinatorial proof of the symmetry of Schur functions. Moreover, we argue that a generalization of Thomas' approach is a natural starting point in search of “universal weighted symmetric functions.”

Sur quelques propriétés de symétrie des nombres de Genocchi

In [2] Dumont stated several conjectures about some symmetric polynomial sequences which are the refinements of the Genocchi numbers. In this paper we shall prove all of his conjectures. We first show that some special cases of his main conjecture can be readily derived from a result of Wall and then give a complete proof of this conjecture by computing some Hankel determinants. Finally, we present a new symmetric model for the Dumont-Foata polynomials in terms of Motzkin paths.