On Some Problems Related to the Differential-Difference Recurrence of Cauchy-Euler Type

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基本資料

Project title

Code/計畫編號

MOST109-2115-M019-001

Translated Name/計畫中文名

Cauchy-Euler 型態的微分─差分型遞迴關係相關問題的分析

Project Coordinator/計畫主持人

Hua-Huai Chern

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Computer Science and Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=13531953

Year

2020

Start date/計畫起

01-08-2020

Expected Completion/計畫迄

31-07-2021

Bugetid/研究經費

272千元

ResearchField/研究領域

數學

筆者擬針對兩類由微分-差分型遞迴關係方程 Differential-Difference Recurrences (DDR) 細分出來的型態進行探究。第一個主體目標為 Recurrence of Cauchy-Euler type (RCE)\[e_n P_n(v) =\sum_{0\le j\le \ell} (1-v)^j \frac{\dd{}^j}{\dd{v}^j}P_{n-1}(v)\sum_{0\le i\le r-j}\!\! d_{j,i}(v)\; n^i,\quad 1\le \ell \le r.\]雖在文獻上能對應的範例不算普遍，但型態上可類比於 ODE 裡的 Cauchy-Euler 方程，同時也缺乏已知的結果，於是將此列為獨立探究的主題之一。另一個主題則專注在不再限制滿足 $(1-v)\mid b_n (v)$ 條件下的 DDR 型態上\[e_n P_n(v)=\big( a_n(v)+c_n(v)\big)P_{n-1}(v)+b_n(v) P_{n-1}'(v),\]其中 $a_n, b_n, c_n$ 等多為 $n,v$ 的多項式而 $e_n$ 為 $n$ 的多項式。後一類型對應的範例相對地較多且多有具體的應用可列舉；然而在缺乏 $(1-v)\mid b_n (v)$ 條件後，少了以階矩分析為主的初等分析工具，如何以 PDE 求解論述來完成後續分析，即為此主題的重點。兩個主體實為筆者過去以廣義 DDR 型式的遞迴關係為研究對象的一系列計畫的分支。即便些許單純的類別可透過簡化與代換等步驟來歸類為已經完成的分析的型態來處理，然而，從一般性的面向來看，相關的理論與分析方法，特別是與採用 PDE 求解論述相關的，依舊不算豐富。在陸續從 OEIS 資料庫或其他領域裡的最新研究成果裡蒐集到更多的實際範例下，於是將這兩型態定為此系列進程下一個階段的核心主題來進行探究。(無法顯示的數學式請參考合併在 CM03 檔案裡的 Latex 版）。The author intends to study two types of models that are branches of Differential-Difference Recurrences (DDR). The first model is the Recurrence of Cauchy-Euler type (RCE)\[e_n P_n(v) =\sum_{0\le j\le \ell} (1-v)^j \frac{\dd{}^j}{\dd{v}^j}P_{n-1}(v)\sum_{0\le i\le r-j}\!\! d_{j,i}(v)\; n^i,\quad 1\le \ell \le r.\]Although the concrete examples associated to this type are not so common, and known results are also rare, the author decides to take it to serve as our first issue. The second part is to investigate the following type of DDR not satisfying $(1-v)\mid b_n(v)$\[e_n P_n(v)=\big( a_n(v)+c_n(v)\big)P_{n-1}(v)+b_n(v) P_{n-1}'(v),\]whose coefficients $a_n, b_n, c_n$ are polynomials of $n,v$ and $e_n$ is polynomial of $n$. For this type, we have more examples and applications. Since the condition $(1-v)\mid b_n(v)$ is not valid, we are no longer able to use elementary approach which is based on the moment analysis to do analysis. Instead, the PDE approach is applicable and becomes essential.Despite that these types are the extensions of current project, and some of the cases can be resolved by using the approaches we have developed ever, but we aim to analyze them since it is more technically involved and only few concrete examples are found. By exploring more examples from new results and OEIS, we are willing to investigate and analyze these models and make them to serve as the core of the next stage of the whole project on DDR.

Keyword(s)

Cauchy-Euler 型遞迴關係
微分─差分型遞迴關係
偏微分方程
畸點
畸點分析
階矩分析
漸進遞移
極限律
Recurrence of Cauchy-Euler type
Differential-Difference Recurrences
partial differential equations
singularity
singularity analysis
moment analysis
asymptotic transfers
limit laws

DSpace CRIS

On Some Problems Related to the Differential-Difference Recurrence of Cauchy-Euler Type

基本資料

Description