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Register a new userOn a family of Jacobi type polynomials as eigenfunctions of $2times 2$ hypergeometric operators: Structural formulas
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Mirta Mar\\'ia Castro Smirnova
Publication date
2021
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English
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We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $2times 2$ introduced by C. Calderon et al. in arXiv:1810.08560, 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 algebra of second order differential operators associated with the weight matrix.
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The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appells function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
In this note, we frst consider boundedness properties of a family of operators generalizing the Hilbert operator in the upper triangle case. In the diagonal case, we give the exact norm of these operators under some restrictions on the parameters. We secondly consider boundedness properties of a family of positive Bergman-type operators of the upper-half plane. We give necessary and sufficient conditions on the parameters under which these operators are bounded in the upper triangle case.
Let ${bf P}_k^{(alpha, beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality begin{equation*} max_{x in [delta_{-1},delta_1]}sqrt{(x- delta_{-1})(delta_1-x)} (1-x)^{alpha}(1+x)^{beta} ({bf P}_{k}^{(alpha, beta)} (x))^2 < frac{3 sqrt{5}}{5}, end{equation*} where $delta_{-1}<delta_1$ are appropriate approximations to the extreme zeros of ${bf P}_k^{(alpha, beta)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erd{e}lyi, A.P. Magnus and P. Nevai conjecture [Erd{e}lyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that begin{equation*} max_{x in [-1,1]}(1-x)^{alpha+{1/2}}(1+x)^{beta+{1/2}}({bf P}_k^{(alpha, beta)} (x))^2 < 3 alpha^{1/3} (1+ frac{alpha}{k})^{1/6}, end{equation*} in the region $k ge 6, alpha, beta ge frac{1+ sqrt{2}}{4}.$
The aim of this work is to demonstrate various an interesting recursion formulas, differential and integral operators, integration formulas, and infinite summation for each of Horns hypergeometric functions $mathrm{H}_{1}$, $mathrm{H}_{2}$, $mathrm{H}_{3}$, $mathrm{H}_{4}$, $mathrm{H}_{5}$, $mathrm{H}_{6}$ and $mathrm{H}_{7}$ by the contiguous relations of Horns hypergeometric series. Some interesting different cases of our main consequences are additionally constructed.
Inspired by the recent work Sahin and Agha gave recursion formulas for $mathcal{G}_{1}$ and $mathcal{G}_{2}$ Horn hypergeometric functions cite{saa}. The object of work is to establish several new recursion relations, relevant differential recursion formulas, new integral operators, infinite summations and interesting results for Horns hypergeometric functions $mathcal{G}_{1}$, $mathcal{G}_{2}$ and $mathcal{G}_{3}$.
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