Related papers: Confluent functions, Laguerre polynomials and thei…
Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…
Let $(P_n(x;z;\lambda))_{n\geq 0}$ be the sequence of monic orthogonal polynomials with respect to the symmetric linear functional $\mathbf{s}$ defined by $$\langle\mathbf{s},p\rangle=\int_{-1}^1 p(x)(1-x^2)^{(\lambda-1/2)}…
In this paper we generalize and specialize generating functions for classical orthogonal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer…
In this paper, we investigated properties of Tribonacci-Lucas polynomials which generalized Tribonacci-Lucas numbers. From this generalization, we also obtain some new algebraic properties on these numbers and polynomials as Binet formula,…
For any complex parameters $x$ and $\nu$, we provide a new class of linear inversion formulas $T = A(x,\nu) \cdot S \Leftrightarrow S = B(x,\nu) \cdot T$ between sequences $S = (S_n)_{n \in \mathbb{N}^*}$ and $T = (T_n)_{n \in…
A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity…
We point out a simple criterion for convergence of polynomials to a concrete entire function in the Laguerre-P\'{o}lya ($\mathcal{LP}$) class (of all functions arising as uniform limits of polynomials with only real roots). We then use this…
In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of…
Multivariable generalizations of the classical Hermite, Laguerre and Jacobi polynomials occur as the polynomial part of the eigenfunctions of certain Schr\"odinger operators for Calogero-Sutherland-type quantum systems. For the generalized…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
Coefficients in the expansions of the form $\partial P_{n}(\lambda;z)}/\partial\lambda=\sum_{k=0}^{n}a_{nk}(\lambda)P_{k}(\lambda;z)$, where $P_{n}(\lambda;z)$ is the $n$th classical (the generalized Laguerre, Gegenbauer or Jacobi)…
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
Several integrals involving powers and ordinary hypergeometric functions are rederived by means of a generalized hypergeometric function of two variables (Appell's function) recovering some well-known expressions as particular cases. Simple…
We study the problem of generalization of Oresme numbers with a new sequence of numbers called Oresme polynomials. Moreover, by using the matrix methods for Oresme polynomials, we obtain the identities including the general bilinear…
We consider the Lommel functions $s_{\mu,\nu}(z)$ for different values of the parameters $(\mu,\nu)$. We show that if $(\mu,\nu)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials…
Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are…
By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms…
The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of…
Laguerre's theorem regarding the number of non-real zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero…
Previous analyses of Laguerre's method have provided results on the convergence and properties of this popular method when applied to the polynomials $p_n(z)=z^n-1$, $n\in\mathbb{N}$. While these analyses appear to provide a fairly complete…