Asymptotics of the generalized Gegenbauer functions of fractional degree
The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by rG (λ) ν (x) (right GGF-Fs) and lG (λ) ν (x) (left GGF-Fs) with x ∈ (−1, 1), λ > −1/2 and real ν ≥ 0, are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of t...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2022
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| Online Access: | https://hdl.handle.net/10356/154539 |
| _version_ | 1826114745338429440 |
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| author | Liu, Wenjie Wang, Li-Lian |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Liu, Wenjie Wang, Li-Lian |
| author_sort | Liu, Wenjie |
| collection | NTU |
| description | The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by rG
(λ)
ν (x) (right
GGF-Fs) and lG
(λ)
ν (x) (left GGF-Fs) with x ∈ (−1, 1), λ > −1/2 and real ν ≥ 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. |
| first_indexed | 2024-10-01T03:44:10Z |
| format | Journal Article |
| id | ntu-10356/154539 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:44:10Z |
| publishDate | 2022 |
| record_format | dspace |
| spelling | ntu-10356/1545392022-01-14T05:52:01Z Asymptotics of the generalized Gegenbauer functions of fractional degree Liu, Wenjie Wang, Li-Lian School of Physical and Mathematical Sciences Science::Mathematics Generalized Gegenbauer Functions of Fractional Degree Asymptotic Analysis The generalized Gegenbauer functions of fractional degree (GGF-Fs), denoted by rG (λ) ν (x) (right GGF-Fs) and lG (λ) ν (x) (left GGF-Fs) with x ∈ (−1, 1), λ > −1/2 and real ν ≥ 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. Ministry of Education (MOE) The research of the first author was supported by the China Postdoctoral Science Foundation Funded Project (No. 2017M620113), the National Natural Science Foundation of China (Nos. 11801120, 71773024 and 11271100), the Fundamental Research Funds for the Central Universities, China (Grant No. HIT.NSRIF.2019058 and No.HIT.NSRIF.2020081) and the Natural Science Foundation of Heilongjiang Province of China (Nos. A2016003 and G2018006). The research of the second author is partially supported by Singapore MOE AcRF Tier 2 Grants: MOE2018-T2-1-059 and MOE2017-T2-2-144. 2022-01-14T05:51:11Z 2022-01-14T05:51:11Z 2020 Journal Article Liu, W. & Wang, L. (2020). Asymptotics of the generalized Gegenbauer functions of fractional degree. Journal of Approximation Theory, 253, 105378-. https://dx.doi.org/10.1016/j.jat.2020.105378 0021-9045 https://hdl.handle.net/10356/154539 10.1016/j.jat.2020.105378 2-s2.0-85079389593 253 105378 en MOE2018-T2-1-059 MOE2017-T2-2-144 Journal of Approximation Theory © 2020 Elsevier Inc. All rights reserved. |
| spellingShingle | Science::Mathematics Generalized Gegenbauer Functions of Fractional Degree Asymptotic Analysis Liu, Wenjie Wang, Li-Lian Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title | Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title_full | Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title_fullStr | Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title_full_unstemmed | Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title_short | Asymptotics of the generalized Gegenbauer functions of fractional degree |
| title_sort | asymptotics of the generalized gegenbauer functions of fractional degree |
| topic | Science::Mathematics Generalized Gegenbauer Functions of Fractional Degree Asymptotic Analysis |
| url | https://hdl.handle.net/10356/154539 |
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