วิยุตเชิงเส้นและสังวัตนาการเชิงการจัด

Kumjorn   Muneekaew; Zithimawor Bunma

Authors

Kumjorn Muneekaew Program in Educational Measurement and Teaching Mathematics, Department of Education, Bansomdejchaopraya Rajabhat University
Zithimawor Bunma -

Keywords:

Discrete linear, Combinatorial convolutions , Linear recurrence relation

Abstract

This article present the discrete linear and combinatorial convolutions with the purpose of explaining the sum of generalized arithmetic-geomtric series that corresponds to the linear recurrence relation and combinatorial convolutions can be summarized as follows

The sum of generalized arithmetic-geomtric series is given by the formula:

$S_{n}(z)=\sum_{j=1}^{n}j^{n}z^{j},z\in C,\left|z\right|<1,n=0,1,2,..$ and $S_{0}(z)=\frac{z}{1-z}$

which corresponds to the recurrence relation and combinatorial convolutions as follows

$S_{n+1}(z)=S_{n}(z)+\sum_{k=0}^{n}\left(\begin{matrix}n\\k\end{matrix}\right)S_{k}(z)S_{n-k}(z),n=0,1,2,...$ and $S_{n}\left(z\right)=\frac{z}{1-z}\left[1+\sum_{k=0}^{n-1}\begin{pmatrix}n\\k\end{pmatrix}S_{k}\left(z\right)\right],n=1,2,...$

References

กำจร มุณีแก้ว. (2563). ผลบวกของอนุกรมเลขคณิต-เรขาคณิตที่วางนัยทั่วไป. วารสารก้าวทันโลกวิทยาศาสตร์. คณะวิทยาศาสตร์และเทคโนโลยี, 19(2), 1-11.

Cîrnu MI. (2011). Determinantal formulas for sum of generalized arithmetic-geometric series. De La Asociacion Matematica Venezolana, 18(1), 25-38.

Kumaresan, S. (2020). Cauchy-Mertens. Retrieved from https://www.youtube.com/Cauchy-Mertens/. Whitman College. (2024). Newton’s Binomial Theorem. Retrieved from https://www.whitman. edu/Newton’ s Binomial Theorem/.

Wikipedia. (2024). Mertens’ Theorem. Retrieved from https://en.wikipedia.org/Mertens’ Theorem/.

DISCRETE LINEAR AND COMBINATORIAL CONVOLUTIONS

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