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Discrete orthogonal polynomials (DOPs) have gained significant research attention owing to their crucial role in digital signal processing applications such as computer vision, pattern recognition, and compression. However, the computation of DOP coefficients often incurs a substantial computational burden, exacerbating for higher-degree moments along with the resulting numerical errors. To address this challenge, this paper exploits the inherent parallelism in Charlier polynomial coefficient calculations to achieve enhanced polynomial performance. Independent calculations are distributed among threads, making efficient use of the available processing resources. Two algorithms are presented, the first algorithm evenly distributes the rows in a sequential manner (straightforward). Additionally, to achieve a more equitable distribution of coefficient calculations, this paper proposes alternative distribution approaches, aimed at balancing processing load among threads. Through extensive comparative experiments, we have confirmed that the proposed approaches achieved high performance across different degrees (1540 to 7370) and at different numbers of threads (2 to 256). The results show processing time in the multithreaded case is improved by up to 9.1 times with respect to the unthreaded case. Furthermore, by increasing the number of threads from 2 to 256, the trend indicates that the most significant improvement occurs in the range of 32 to 128 threads, confirming the robustness of the proposed algorithm. These findings signify the importance of this paper.

The discrete Charlier moment transform, while extensively utilized in image processing, is inherently lossy and non-integer reversible, making it unsuitable for lossless image applications. To address this, we propose the Integer Reversible Charlier Transform (IRCT), which operates on integer values and produces integer coefficients, enabling perfect and unique recovery of the original input data. The IRCT maintains the orthogonality and invertibility ensuring exact similarity between original and reconstructed images. We leverage the capabilities of the IRCT to develop a novel reversible data hiding (RDH) scheme. This scheme embeds additional data into images by modifying the histogram in the IRCT domain. By capitalizing on the concentrated nature of the IRCT histogram, characterized by high peaks, our method achieves a significantly high embedding capacity while preserving image quality and robustness to statistical attacks. Comparative performance evaluations underscore the effectiveness of the IRCT-based RDH scheme over existing techniques across various domains, positioning it as a promising solution for secure data transmission.

We give new explicit formulas as well as new generating functions for the associated Meixner, Charlier, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this paper is to define and investigate more general multiple Charlier polynomials on the linear lattice ω N = { 0 , ω , 2 ω , … } $\omega \mathbb{N} = \{ 0,\omega ,2\omega ,\ldots \} $ , ω ∈ R $\omega \in \mathbb{R}$ . We call these polynomials ω-multiple Charlier polynomials. Some of their properties, such as the raising operator, the Rodrigues formula, an explicit representation and a generating function are obtained. Also an ( r + 1 ) $( r+1 )$ th order difference equation is given. As an example we consider the case ω = 3 2 $\omega =\frac{3}{2}$ and define 3 2 $\frac{3}{2}$ -multiple Charlier polynomials. It is also mentioned that, in the case ω = 1 $\omega =1$ , the obtained results coincide with the existing results of multiple Charlier polynomials.

Abstract In this paper, we give some direct approximation results by modified Kantorovich–Szász type operators involving Charlier polynomials. Further, approximation results are also developed in polynomial weighted spaces. Moreover, for the functions of bounded variation, approximation results are proved. Finally, some graphical examples are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions.

This paper discusses the applicability of data reconciliation approaches in metrology and mentions the existed shortcomings. The semi-parametric method based on Gram-Charlier series expansion is presented for overcoming the obstacles preventing the wider spread of the measured data reconciliation in metrological practice. The proposed approach allows the fast estimation of potential accuracy increase that matters for adaptive measurement systems. The corresponded expressions and tests are presented.