本論文主要是針對多變數的拉格朗日多項式(Multivariable Lagrange polynomials)方面所作之研究。
第一章:緒論。
第二章:導出多變數的拉格朗日多項式相關的一些等式,並且在雙變數的情況下,可經由一些轉換即可推廣成,相關文獻中所研究(雅可比多項式、拉蓋爾多項式、超幾何多項式)的主要等式。另外也研究一些線性偏微分算子與多變數的拉格朗日多項式之間的關係。
第三章:將三個變數的拉格朗日多項式與Appell函數的雙側生成函數(Bilateral generating functions),推廣成多變數的拉格朗日多項式與Lauricella函數的雙側生成函數。
第四章:藉由變數變換再取極限,去導出新的多項式,並求出對應於拉格朗日多項式的一些等式和遞迴關係。
第五章:利用Bailey 三次轉換去導出雙重級數的等式,並藉此去求出
Srivastava-Daoust 的轉換公式與歸約公式。

The main purpose of this thesis is to investigate multivariable Lagrange polynomials.
In Chapter 1, introduction.
In Chapter 2, we derive some identities of Lagrange polynomials of two variables and observe the relations between Lagrange polynomials and Jacobi, Laguerre
and Hypergeometric polynomials, respectively. On the other hand, we investigate linear partial differential operators on multivariable Lagrange polynomials.
In Chapter 3, we generalize bilateral generating functions for the Lagrange polynomials with three variables and the Appell functions, as bilateral generating functions for the Lagrange polynomials with multivariable and the Lauricella functions.
In Chapter 4, using the method of substitution and taking limit, we obtain some new polynomials. We obtain some identities and recurrence relations.
In Chapter 5, Based upon Bailey’s cubic transformations, we construct some identities and use item to find transformation and reduction formulas for the
Srivastava-Daoust hypergeometric function in two variables.

1 緒論1
2 拉格朗日多項式(Lagrange polynomials) 3
2.1 符號介紹(Notations) . . . . . . . . . . . . . 3
2.2 雙變數的拉格朗日多項式(Two-variable Lagrange polynomials) . . 4
2.2.1 雅可比多項式(Jacobi polynomials) 與雙變數的拉格朗日
多項式之間的關係. . . . . . . . . . . . . . . . . . 4
2.2.2 拉蓋爾多項式(Laguerre polynomials) 與雙變數的拉格朗
日多項式之間的關係. . . . . . . . . . . . . . . . . 6
2.2.3 超幾何多項式(Hypergeometric polynomials) 與雙變數的拉
格朗日多項式之間的關係. . . . . . . . . . . . . . . 7
2.3 多變數的拉格朗日多項式(Multivariable Lagrange polynomials) 的定義與性質 . . . . . . . . . . . 8
2.4 多變數的拉格朗日多項式相關的等式與生成函數. . . 11
2.5 雅可比多項式相關的等式. . . . . . . . . . . . . 19
2.6 拉蓋爾多項式相關的等式. . . . . . . . . . . . . 24
2.7 超幾何多項式相關的等式. . . . . . . . . . . . . 26
2.8 多變數的拉格朗日多項式與線性偏微分算子(Linear partial
differential operators) 之間的關係. . . . . . . . . 27
2.9 由第二類斯特林數(Stirling numbers of the second kind) 所得到的等式之推廣. . . . . . . . . . . . . . . . . . 33

3 雙側生成函數(Bilateral generating functions) 35
3.1 介紹. . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Appell 函數(Appell functions) . . . . . . . . 35
3.1.2 Lauricella 函數(Lauricella functions) . . . . 37
3.1.3 四重超幾何函數(Quadruple hypergeometric functions) . . . . 38
3.1.4 多重超幾何函數(Multiple hypergeometric functions) . . . . . . 39
3.1.5 Kamp´e de F´eriet 函數(Kamp´e de F´eriet functions) . . . . . . . . 40
3.1.6 廣義的Lauricella 函數(Generalized Lauricella function) . . . . . 40
3.2 多變數的拉格朗日多項式與Lauricella 函數的雙側生成函數41
3.3 多變數拉格朗日多項式的多重加法公式(multiple addition formula). . . . . . . 48
4 拉格朗日多項式所產生的新多項式50
4.1 f_k (x1, ..., xr)的定義與性質. . . . . . . . . 50
4.2 f_k (x1, ..., xr)所滿足的偏微分方程式與遞迴關係. . 51
4.3 f_k (x1, ..., xr)的多重加法公式(multiple addition formula) . . 53
5 一些二重級數的等式和連帶(相伴) 生成函數的關係(Some double-series identities and associated generating-function
relationships) 55
5.1 符號介紹與定義(Introduction and definitions) . . . 57
5.2 利用Bailey''s 三次轉換所導出之級數的等式(Series identities based upon Bailey''s cubic transformations) . 57
5.3 可約二重超幾何函數(Reducible double hypergeometric functions) . 61
5.4 連帶(相伴) 生成函數的關係(Associated generating-function relationships). . . . . . . . 63
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