Hypergeometric Operators Research Articles

SECOND ORDER DIFFERENTIAL OPERATORS WITH ALGEBRAIC SOLUTIONS

We are surveying recent results that describe second order differential operators having only algebraic solutions in the sense of Galois theory. We call such operators algebraic. For hypergeometric operators, this problem was studied by Schwarz and Klein who also gave results that describe all second order linear differential operators with a full set of algebraic solutions. Starting from their work, we see algebraic operators as pull-backs of algebraic hypergeometric operators via Belyi functions. We are surveying some of the main results describing second order operators with a full set of algebraic solutions, especially those obtained by using the properties of the pull-back functions. Using the Grothendieck correspondence, these properties transfer to properties for their corresponding dessins d’enfants.

An asymptotic expansion of eigenpolynomials for a class of linear differential operators

AbstractConsider an M‐th order linear differential operator, ,where is a monic complex polynomial such that and are complex polynomials such that . It is known that the zero counting measure of its eigenpolynomials converges in the weak star sense to a measure μ. We obtain an asymptotic expansion of the eigenpolynomials of in compact subsets out of the support of μ. In particular, we solve a conjecture posed in Masson and Shapiro [On polynomial eigenfunctions of a hypergeometric type operator. Exper Math. 2001;10:609‐618].

Composition Formula for Saigo Fractional Integral Operator Associated with V‐Function

In this study, we form integral formulas for Saigo’s hypergeometric integral operator involving V‐function. Corresponding assertions for the classical Riemann–Liouville (R‐L) and Erdélyi–Kober (E‐K) fractional integral operator are extrapolated. Also, by putting in the transformations of Beta and Laplace, we can establish their composition formulas. By selecting the appropriate parameter values, the V‐function may be reduced to a variety of functions, including the exponential function, Mittag–Leffler, Lommel, Struve, Wright’s generalized Bessel function, and Bessel and generalized hypergeometric function.

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to 2times 2 Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2times 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

Fuzzy Differential Subordinations Obtained Using a Hypergeometric Integral Operator

This paper is related to notions adapted from fuzzy set theory to the field of complex analysis, namely fuzzy differential subordinations. Using the ideas specific to geometric function theory from the field of complex analysis, fuzzy differential subordination results are obtained using a new integral operator introduced in this paper using the well-known confluent hypergeometric function, also known as the Kummer hypergeometric function. The new hypergeometric integral operator is defined by choosing particular parameters, having as inspiration the operator studied by Miller, Mocanu and Reade in 1978. Theorems are stated and proved, which give corollary conditions such that the newly-defined integral operator is starlike, convex and close-to-convex, respectively. The example given at the end of the paper proves the applicability of the obtained results.

A pair of commuting hypergeometric operators on the complex plane and bispectrality

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.

Time–frequency analysis associated with the generalized Wigner transform

Time–frequency analysis theory has relatively recent developments in pure and applied mathematics. Motivated by Wong's approach, we will study, in this paper, the localization operators associated with the generalized Wigner transform. Results on the -boundedness and -compactness of these localization operators are given. The proofs are based on some estimates on the generalized hypergeometric translation operators, the generalized Wigner transform and interpolation theory.

Fractional calculus of a product of multivariable Srivastava polynomial and multi-index Bessel function in the kernel <i>F</i><sub>3</sub>

In this article our main object to compute image formulas of generalized fractional hypergeometric operators, involving the product of multivariable Srivastava polynomial and multiindex Bessel function. The results obtained provide unification and an extension of known results given earlier by Agarwal and Nieto [ 1 ], Agarwal et al. [ 2 ] Mishra et al. [ 18 ], Saxena and Saigo [ 26 ], Suthar et al. [ 32 ]. We also consider certain special cases of derived results by specializing suitable value of the parameters.

2 × 2 hypergeometric operators with diagonal eigenvalues

In this work we give all the order-two hypergeometric operators D, symmetric with respect to some 2 × 2 irreducible matrix-weight W on (0,1) such that DPn=Pnλn00μn with no repetition among the eigenvalues {λn,μn}n∈N0, where {Pn}n∈N0 is the (unique) sequence of monic orthogonal polynomials with respect to W.We obtain a new family of such operators and weights, depending on three parameters, generalizing some older examples. We also give, in a very explicit way, the corresponding monic orthogonal polynomials, their three term recurrence relation and their squared matrix-norms.

Quantum mirror curve of periodic chain geometry

The mirror curves enable us to study B-model topological strings on noncompact toric Calabi-Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with a single brane. In this paper, we discuss two types of geometries: one is the chain of N ℙ1’s which we call “N-chain geometry,” the other is the chain of N ℙ1’s with a compactification which we call “periodic N-chain geometry.” We calculate the partition functions of the open topological strings on these geometries, and obtain the mirror curves and their quantization, which is characterized by (elliptic) hypergeometric difference operator. We also find a relation between the periodic chain and ∞-chain geometries, which implies a possible connection between 5d and 6d gauge theories in the larte N limit.

The harmonic analysis associated to the Heckman-Opdam's theory and its application to a root system of type $BC_d$

In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and ${}^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on $\mathbb{R}^d$. By using these operators we define the hypergeometric translation operator $\mathcal{T}^W_x, x \in \mathbb{R}^d$, and its dual ${}^t\mathcal{T}^W_x, x \in \mathbb{R}^d$, we express them in terms of the hypergeometric Fourier transform $\mathcal{H}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform $\mathcal{H}^W$. We study also the hypergeometric convolution product on $W$-invariant $L^p_{\mathcal{A}_k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

Tridiagonalization and the Heun equation

It is shown that the tridiagonalization of the hypergeometric operator L yields the generic Heun operator M. The algebra generated by the operators L, M and Z = [L, M] is quadratic and a one-parameter generalization of the Racah algebra. The Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators L and M. An interpretation in terms of the Racah problem for su(1,1) and separation of variables in a superintegrable system are discussed.

The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman--Opdam theory on ℝ d

Abstract We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman-Opdam theory on

Operator on Hilbert space and its application to certain multivalent functions with fixed point associated with hypergeometric function

By applying hypergeometric operator on Hilbert space, the author introduces a new class of meromorphic multivalent functions with an arbitrary fixed point omega. Properties such as coefficient inequalities, distortion bounds and extreme points were derived. Furthermore, the effect of this operator on functions in this class was also investigated.

Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra

The algebraic underpinning of the tridiagonalization procedure is investigated. The focus is put on the tridiagonalization of the hypergeometric operator and its associated quadratic Jacobi algebra. It is shown that under tridiagonalization, the quadratic Jacobi algebra becomes the quadratic Racah–Wilson algebra associated to the generic Racah–Wilson polynomials. A degenerate case leading to the Hahn algebra is also discussed.

Certain recent fractional integral inequalities associated with the hypergeometric operators

The principle aim of this paper is to establish some new (presumably) fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Riemann–Liouville type fractional integral operators by using hypergeometric fractional integral operator. Some relevant connections of the results presented here with those earlier ones are also pointed out.

ABSOLUTE CONTINUITY OF THE REPRESENTING MEASURES OF THE HYPERGEOMETRIC TRANSLATION OPERATORS ATTACHED TO THE ROOT SYSTEM OF TYPE B2AND C2

We prove in this paper the absolute continuity of the representing measures of the hypergeometric translation operators $\mathcal{ T}_ x$ and $\mathcal{ T}_ x^ W$ associated respectively to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type $ B_ 2$ and $ C_ 2$ which are studied in [9].

Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere

In this paper, we describe the irreducible spherical functions of fundamental $K$-types associated with the pair $(G,K)=({\mathrm{SO}}(n+1),{\mathrm{SO}}(n))$ in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same $K$-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size $2$ and $3$, with respect to a matrix-weight $W$ supported on $[0,1]$. Moreover, we show that $W$ has a second order symmetric hypergeometric operator $D$.

THE POSITIVITY OF THE HYPERGEOMETRIC TRANSLATION OPERATORS ASSOCIATED TO THE CHEREDNIK OPERATORS AND THE HECKMAN-OPDAM THEORY ATTACHED TO THE ROOT SYSTEMS OF TYPE B2AND C2

We consider the hypergeometric translation operator associated to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type <TEX>$B_2$</TEX>. We prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular we deduce that the product formulas of the Opdam-Cherednik and the Heckman-Opdam kernels are positive integral transforms, and we obtain best estimates of these kernels. The method used to obtain the previous results shows that these results are also true in the case of the root system of type <TEX>$C_2$</TEX>.

On Fractional Integral Inequalities Involving Hypergeometric Operators

Here we aim at establishing certain new fractional integral inequalities involving the Gauss hypergeometric function for synchronous functions which are related to the Chebyshev functional. Several special cases as fractional integral inequalities involving Saigo, Erdélyi-Kober, and Riemann-Liouville type fractional integral operators are presented in the concluding section. Further, we also consider their relevance with other related known results.