Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1701.07996

Orthogonal Polynomials related to g-fractions with missing terms

Kiran Kumar Behera·A. Swaminathan
View original source ↗Read source document ↗Export metadata ↓

Abstract

The purpose of the present paper is to investigate some structural and qualitative aspects of two different perturbations of the parameters of $g$-fractions. In this context the concept of \emph{gap} $g$-fractions is introduced. While tail sequences of a continued fraction play a significant role in the first perturbation, Schur fractions are used in the second perturbation of the $g$-parameters that are considered. Illustrations are provided using Gaussian hypergeometric functions. Using a particular gap $g$-fraction, some members of the class of Pick functions are also identified.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kiran Kumar Behera, A. Swaminathan. 2017-01-27. Orthogonal Polynomials related to g-fractions with missing terms. https://arxiv.org/abs/1701.07996

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Entropy Expansion for General Polynomial Images of Frostman Random Variables

We prove dyadic entropy expansion for the observables $X+Y$ and $f(X,Y)$ under Frostman nonconcentration hypotheses on a prescribed, possibly dependent law. For an integer $n\ge1$, write $H_n(Z)=H(\lfloor 2^nZ\rfloor)$ for the base-two Shannon entropy at resolution $2^{-n}$; thus $n$ indexes the fineness of the dyadic discretization. For every $0 4/3$, we obtain $\max\{H_n(X+Y),H_n(f(X,Y))\}\ge (\frac{s_1+s_2}{2}+\varepsilon)n-O(1)$ for an explicit $\varepsilon>0$. Here the baseline averages the two Frostman exponents. We also classify the exceptional coordinate representations and construct obstructions for algebraic affine directions with Frostman constants uniform in the scale.

math.CA↗
arXiv2026

On entire solutions of two kinds of quadratic trinomial Fermat type functional equations

The existence of entire solutions to Fermat type differential-difference equations and \(q\)-difference differential equations involving second-order derivatives is investigated by using Nevanlinna theory. The exact forms of the entire solutions to these equations mentioned above are identified. These results represent a generalization and improvement of previous findings obtained by Gong et al. [Bull. Iran. Math. Soc. 51, 17 (2025)]. Furthermore, some examples are provided to illustrate these results.

math.CA↗
arXiv2026

On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

Let $p$ be an odd prime. We prove the extension estimate $R_{S_j}^*(2\to r)\lesssim_r 1$ for every nonzero-radius sphere $S_j\subseteq\mathbb{F}_p^4$ and every $r\geq 16/5$, uniformly in $p$ and $j$. This improves the Stein--Tomas exponent $10/3$ established by Iosevich and Koh (2008). To prove this result, we develop a new method that combines horizontal slicing, cancellation in an affine fourth-moment expansion, a rich-plane decomposition, and incidence estimates. We also formulate a localized spherical restriction/extension conjecture that predicts the sharp dependence of the restriction norm on the size of the physical support. This conjecture implies the spherical extension estimates $R_{S_j}^*(2\to r)\lesssim_r 1$ for every $r>3$, and yields almost-every-pin distance estimates at the conjectured Erdős--Falconer exponent in four dimensions, up to an arbitrarily small power loss in the set-size hypothesis. Using the same method, we improve the bounds supplied by Fourier decay and Plancherel at intermediate support scales and derive new almost-every-pin distance estimates in $\mathbb{F}_p^4$.

math.CA↗