Non-classical polynomials and the inverse theorem

In this paper we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$ -norm must correlate with a classical polynomial when $k\le p+1$ . To the best of our knowledge, this result is new for $k=p+1$ (when $p>2$ ). We then prove that non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$ -norm over $\mathbb F_p^n$ for all $k\ge p+2$ , completely characterising when classical polynomials suffice.

Some Generalized Results to unify Classical Polynomials

Present work of this paper deals with the unification of classical polynomials in which we have defined a generalized polynomial set analogous to that of associated Legendre polynomial P (x)mn by taking 5the use of Operator. Also we have derived explicit form, OperationalFormulae generating functions for this function.

ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

The derivative connecting problems for some classical polynomials

Given two polynomial sets $\{ P_n(x) \}_{n\geq 0},$ and $\{ Q_n(x) \}_{n\geq 0}$ such that $$\deg ( P_n(x) ) =n, \deg ( Q_n(x) )=n.$$ The so-called the connecting problem between them asks to find the coefficients $\alpha_{n,k}$ in the expression $\displaystyle Q_n(x) =\sum_{k=0}^{n} \alpha_{n,k} P_k(x).$ Let $\{ S_n(x) \}_{n\geq 0}$ be another polynomial set with $\deg ( S_n(x) )=n.$ The general connection problem between them consists in finding the coefficients $\alpha^{(n)}_{i,j}$ in the expansion $$Q_n(x) =\sum_{i,j=0}^{n} \alpha^{(n)}_{i,j} P_i(x) S_{j}(x).$$ The connection problem for different types of polynomials has a long history, and it is still of interest. The connection coefficients play an important role in many problems in pure and applied mathematics, especially in combinatorics, mathematical physics and quantum chemical applications. For the particular case $Q_n(x)=P'_{n+1}(x)$ the connection problem is called the derivative connecting problem and the general derivative connecting problem associated to $\{ P_n(x) \}_{n\geq 0}.$ In this paper, we give a closed-form expression of the derivative connecting problems for well-known systems of polynomials.

Convergence and Extended Linear Generating Function for Generalized Hypergeometric Function

The subject of Special functions has a lot importance during the last few decades. The intend of this work is to test the convergence and to introduce the extended linear generating relation for the generalized hypergeometric function. The result is followed by its applications to the classical polynomials.

POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

Wavelet Solution for Biosensor Response at Mixed Enzyme Kinetics

This paper presents the approximate solution of the reaction diffusion equation based on the hybridization of classical polynomials and Legendre wavelets. The systems of equations are generated first for the differential equations by the properties of Legendre wavelets. Theoretical analysis for the proposed scheme is discussed and computed solutions are also compared with other numerical solutions available in the literature.

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.