Uniform bounds and asymptotics of Generalized Gegenbauer functions of fractional degree

TL;DR

This paper establishes uniform bounds and asymptotic behaviors of generalized Gegenbauer functions of fractional degree, enhancing understanding of their properties and applications in polynomial approximation and fractional differential equations.

Contribution

It derives explicit residual expressions and uniform bounds for GGF-Fs, and explores their asymptotics and properties, filling gaps in their theoretical understanding.

Findings

Derived explicit residual term for GGF-Fs.

Established uniform bounds in fractional degree and variable.

Analyzed asymptotic behavior for large fractional degrees.

Abstract

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for $\lambda,\nu>0$ and $x=\cos\theta$ with $\theta\in (0,\pi),$ \begin{equation*}\label{IntRep-0N} (\sin \varphi)^{\lambda}\,{}^{r\!}G_\nu^{(\lambda)}(\cos \varphi)= \frac{2^\lambda\Gamma(\lambda+1/2)}{\sqrt{\pi} {(\nu+\lambda)^{\lambda}}} \, {\cos ((\nu+\lambda)\varphi- \lambda\pi/2)} +{\mathcal R}_\nu^{(\lambda)} (\varphi), \end{equation*} and derive the precise expression of the "residual" term ${\mathcal R}_\nu^{(\lambda)} (\varphi).$ With this at our disposal, we obtain the bounds of GGF-Fs uniform in $\nu.$ Under an appropriate weight function, the bounds are uniform for $\theta\in [0,\pi]$ as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree $\nu.$ The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.