Diego Dominici
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
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Oscillatory integrals with polynomial phase and regularity of distributions
本文研究了多项式相位的振荡积分与分布的正则性，通过获得 $s$-凹测度和乘积测度的多项式像密度连续性模的无量纲估计，解决了 Carbery 和 Wright (2001) 关于凸集上多项式相位振荡积分精确上界的猜想。
Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
For the polynomials orthogonal on the unit circle with respect to the measure from the Szego class we prove that the polynomial entropy integrals can grow. The estimate obtained is sharp.
We recast Byerly's formula for integrals of products of Legendre polynomials. Then we adopt the idea to the case of Jacobi polynomials. After that, we use the formula to derive an asymptotic formula for integrals of products of Jacobi polynomials. The asymptotic formula is similar to an analogous one recently obtained by the first author and Jeff Geronimo for a different case. Thus, it suggests that such an asymptotic behavior is rather generic for integrals of products of orthogonal polynomials.
We generalize generating functions for hypergeometric orthogonal polynomials, namely Jacobi, Gegenbauer, Laguerre, and Wilson polynomials. These generalizations of generating functions are accomplished through series rearrangement using connection relations with one free parameter for these orthogonal polynomials. We also use orthogonality relations to determine corresponding definite integrals.
Motivated by an expression by Persson and Strang on an integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre and Gegenbauer polynomials as well as the original Legendre polynomial with even index.
In this paper, we shall prove the uniform sharp $L^p$ decay estimates for a class of oscillatory integral operators with polynomial phases. By this one-dimensional result, we can use the rotation method to obtain uniform sharp $L^p$ estimates of certain higher-dimensional oscillatory integral operators.
By applying an integral representation for $q^{k^{2}}$ we systematically derive a large number of new Fourier and Mellin transform pairs and establish new integral representations for a variety of $q$-functions and polynomials that naturally arise from combinatorics, analysis, and orthogonal polynomials corresponding to indeterminate moment problems. These functions include $q$-Bessel functions, the Ramanujan function, Stieltjes--Wigert polynomials, $q$-Hermite and $q^{-1}$-Hermite polynomials, and the $q$-exponential functions $e_{q}$, $E_{q}$ and $\mathcal{E}_{q}$. Their representations are …
We consider in this work planar polynomial differential systems having a polynomial first integral. We prove that these systems can be obtained from a linear system through a polynomial change of variables.
We give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity. In the affirmative case, our algorithm computes a minimal first integral. In addition, we solve the Poincaré problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field.
The aim of this paper is two fold. We derive an integral representation for the generalized 2D Zernike polynomials which are of independent interest and give the explicit expression of the action of the Cauchy transform on them.
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 polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generating functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given.
We present an integral product formula for Jack polynomials of two variables, extending that of zonal polynomials. It provides another way to find the explicit integral representation for the generalized Bessel function of type $ B_2 $, as presented in \cite{D}.
We give several expansion and identities involving the Ramanujan function $A_q$ and the Stieltjes--Wigert polynomials. Special values of our idenitities give $m$-versions of some of the items on the Slater list of Rogers-Ramanujan type identities. We also study some bilateral extensions of certain transformations in the theory of basic hypergeometric functions.
We obtain some new properties of the signal generated by the Riemann zeta-function in this paper. Namely, we show the connection between the function $\zf$ and a nonlinear integral equation related to the Poisson-Lobachevsky integral.
A new approach to evaluating Malmsten's integral and related integrals
本文通过引入第一类符号广义Stirling多项式和完全Bell多项式，推广了对数积分和双曲积分，给出了Malmsten积分在双曲正割函数自然幂次下的推广形式，并推导了积分序列的约化公式。
In this paper, we shall prove the $L^{p}$ endpoint decay estimates of oscillatory integral operators with homogeneous polynomial phases $S$ in $\mathbb{R} \times \mathbb{R}$. As a consequence, sharp $L^{p}$ decay estimates are also obtained when polynomial phases have the form $S(x^{m_{1}},y^{m_{2}})$ with $m_1$ and $m_2$ being positive integers.
Confluent functions, Laguerre polynomials and their (generalized) bilinear integrals
本文研究了连函数与拉格朗日多项式的双线性积分性质，揭示了其在不同参数下的正交关系及广义格拉姆矩阵的构建方法。
In the first part, by the first author's work of 1972, an integral representation for an ultraspherical polynomial of higher index in terms of one of lower index and an infinite series was obtained. While this representation works well from a theoretical point of view, it is not numerically satisfactory as it involves polynomials of high degree, which are numerically unstable. Here we sum this series to obtain an integral, which is numerically tractable.
A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight
该研究针对带对数权的勒让德多项式乘积积分问题，给出了阶数为m、n的勒让德多项式对应区间[0,1]上含对数权乘积积分的简单推导方法。
In this paper we establish different representations of the so-called Yor integral, which is one of the key ingredient in mathematical finance, in particular, to compute normalized prices of Asian options. We show, that the Yor integral is related with the Kontorovich-Lebedev transform. Also we discuss its relationship with a system of polynomials recently introduced by the author. We derive new important properties of these polynomials, including upper bounds, an exact asymptotic behavior for large values of their degree and explicit formula of coefficients.
We obtain sharp $L^p$ bounds for oscillatory integral operators with generic homogeneous polynomial phases in several variables. The phases considered in this paper satisfy the rank one condition which is an important notion introduced by Greenleaf, Pramanik and Tang. Under certain additional assumptions, we can establish sharp damping estimates with critical exponents to prove endpoint $L^p$ estimates.
We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein-Chen-Dilcher's polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of iterated integrals of ultraspherical polynomials with parameters $(2α+1)/2$, $α\in \mathbb{Z}_+$.
We prove a generalization of the $q$-Selberg integral evaluation formula. The integrand is that of $q$-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.
We construct 3 finite systems of $4-F-3$ hypergeometric orthogonal polynomials. The weights are 1) the weight defined by the $5-H-5$ Dougall summation formula; 2) the integrand in the Askey beta-integral; 3) the weight $w(s)=|p(s)/q(s)|^2$, where $p(s)=Γ(a+is)Γ(b+is)Γ(c+is)$ and $q(s)=Γ(d+is)Γ(2is)$. We also evaluate the integral of the function $w(s)$ using the Jacobi transform (the Olevsky transform); this integral also can be reduced to the Nassrallah--Rahman integral.
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.
Persson and Strang (2003) evaluated the integral over [-1,1] of a squared odd degree Legendre polynomial divided by x^2 as being equal to 2. We consider a similar integral for orthogonal polynomials with respect to a general even orthogonality measure, with Gegenbauer and Hermite polynomials as explicit special cases. Next, after a quadratic transformation, we are led to the general nonsymmetric case, with Jacobi and Laguerre polynomials as explicit special cases. Examples of indefinite summation also occur in this context. The paper concludes with a generalization of the earlier results for H…
We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of gaussian functions and of multiple products of Hermite polynomials.
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals.
We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integ…
The sharp range of $L^p$-estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positive-definite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments.
In this paper we consider the problem on uniform estimates for generalized oscillatory integrals given by Mittag- Leffler functions with the homogeneous polynomial phase. We obtain a variant of Ricci-Stein Lemma and invariant estimates for corresponding integrals.
Sharp $L^2$ decay rate for (1+2)-dimensional oscillatory integral operators with cubic polynomial phases
本文研究了(1+2)维振荡积分算子中三阶多项式相位的$L^2$衰减率，证明了文献中给出的3/8衰减率是精确的。
In this paper, we consider the degenerate and singular oscillatory integral operator with a singular kernel which is not a Calderón-Zygmund kernel and satisfies suitable size and derivative conditions related to a real parameter $μ$. For any given homogeneous polynomial phases, except monomial phases, of degreee $n$, we give the range of $p$ for which the sharp decay rate $-\frac{1-μ}{n}$ on $L^2$ spaces can be preserved on $L^p$ spaces.
Let $K$ be a standard Hölder continuous Calderón--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.
This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.
We give estimates on discrete fractional integral operators along binary quadratic forms. These operators have been studied for 30 years starting with the investigations of Arkhipov and Oskolkov, but efforts have concentrated on cases where the phase polynomial is translation invariant or quasi-translation invariant. This work presents the first results for operators with neither translation invariant nor quasi-translation invariant phase polynomials.
We give integral representations for multiple Hermite and multiple Hermite polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.
We derive a novel integral representations of Jacobi polynomials in terms of the Gauss hypergeometric function. Such representation is then used to give the explicit integral representation for the Heat kernel on the quantized Riemann sphere.
A sequence of coefficients that appeared in the evaluation of a rational integral has been shown to be unimodal. An alternative proof is presented.
We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties…
It is shown in this paper that the Jacob's ladder is the asymptotic solution to the new nonlinear integral equations which correspond to the functions $J_ν(x)$ and $|\zf|^4$.
It is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the Bessel's functions. In this direction, a new class of the nonlinear integral equations is introduced.
A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals $(u_0,u_1)$ was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties. Integral Transforms Spec Funct. 2021;32(2):134-153]. Six interesting special cases were pointed out there. For each case, we precisely deal with the integral representation problem for the functionals associated to these polynomials. The focus is on the matrix differential equation $\big({\bfΦU}\big)'+{\bfΨU}=0$, with ${\bf U}={^t}(u_0…
We study in detail two families of $q$-Fibonacci polynomials and $q$-Lucas polynomials, which are defined by non-conventional three-term recurrences. They were recently introduced by Cigler and have been then employed by Cigler and Zeng to construct novel $q$-extensions of classical Hermite polynomials. We show that both of these $q$-polynomial families exhibit simple transformation properties with respect to the classical Fourier integral transform.
We construct new families of (q-) difference and (contour) integral operators having nice actions on Koornwinder's multivariate orthogonal polynomials. We further show that the Koornwinder polynomials can be constructed by suitable sequences of these operators applied to the constant polynomial 1, giving the difference-integral representation of the title. Macdonald's conjectures (as proved by van Diejen and Sahi) for the principal specialization and norm follow immediately, as does a Cauchy-type identity of Mimachi.
Integral and hypergeometric representations for multiple orthogonal polynomials
本文为任意权函数数量的经典多重正交多项式，建立了I型Hahn多项式的显式超几何表达式，并利用留数定理和Mellin变换推导了Laguerre、Jacobi-Piñeiro和Hahn族的围道积分表示，进而得到显式超几何表示。
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including parameters, then a holonomic difference-differential system for the integral can also be computed. In the algorithm, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function.
In this paper we investigate the asymptotic behavior of polynomials that are orthonormal over the interior domain of an analytic Jordan curve L with respect to area measure. We prove that, inside L, these polynomials behave asymptotically like a sequence of certain integrals involving the canonical conformal map of the exterior of L onto the exterior of the unit circle and certain meromorphic kernel function defined in terms of a conformal map of the interior of L onto the unit disk. The error term in the integral representation is proven to decay geometrically and sufficiently fast, allowing …
The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the $q$-Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. It is noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in $\hat{sl_2}$-confromal field theory.
In this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y), dY/dt= B(X, Y), where A,B belongs to Z[X, Y], degA \leq d, degB \leq d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y)dX + B(X, Y)dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential num…
We prove a set of general theorems that provide new nonlocal constants and first integrals for nonlinear Jacobi-type ordinary differential equations. Applications include equations of the Painleve-Gambier classification.
We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\bigOsoft(N^{2 ω})$ and to a deterministic algorithm solving the problem in $\bigOsoft(d^2N^{2 ω+1})$ arithmetic operations, where $N$ denotes the given bound for the degree of the ration…
Classical discrete multiple orthogonal polynomials: hypergeometric and integral representations
本文研究任意权重的经典离散多重正交多项式，推导了Hahn多项式递归系数的显式表达式，利用多重Askey方案为各后裔族提供了I型多项式及递归系数的超几何与积分表示，并补充了相应的极限以完善该方案。
Using the Baker-Akhiezer function technique we construct a separation of variables for the classical trigonometric 3-particle Ruijsenaars model (relativistic generalization of Calogero-Moser-Sutherland model). In the quantum case, an integral operator M is constructed from the Askey-Wilson contour integral. The operator M transforms the eigenfunctions of the commuting Hamiltonians (Macdonald polynomials for the root sytem A2) into the factorized form S(y1)S(y2) where S(y) is a Laurent polynomial of one variable expressed in terms of the 3phi2(y) basic hypergeometric series. The inversion of M …
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ism…
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