Result: q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials: \(q\)-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

Title:
q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials: \(q\)-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials
Authors:
Zeinab S. Mansour, Mourad E. H. Ismail
Source:
Analysis and Applications. 17:853-895
Publisher Information:
World Scientific Pub Co Pte Lt, 2019.
Publication Year:
2019
Subject Terms:
\(q\)-analog of Bernoulli polynomials, Difference equations, scaling (\(q\)-differences), basic hypergeometric series, 01 natural sciences, Power series (including lacunary series) in one complex variable, two-point \(q\)-Taylor series, \(q\)-analog of Lidstone expansion theorem, Basic hypergeometric functions in one variable, \({}_r\phi_s\), Cauchy kernel, \(q\)-calculus and related topics, 8. Economic growth, 0101 mathematics, Bernoulli and Euler numbers and polynomials, Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
Document Type:
Academic journal Article
File Description:
application/xml
Language:
English
ISSN:
1793-6861
0219-5305
DOI:
10.1142/s0219530518500264
Access URL:
https://zbmath.org/7115545
https://doi.org/10.1142/s0219530518500264
https://www.worldscientific.com/doi/abs/10.1142/S0219530518500264
Accession Number:
edsair.doi.dedup.....bdacac566487e83cd88ef17d489f63ee
Database:
OpenAIRE

Further Information

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].