本文選題：Gamma函數(shù)　切入點：ψ函數(shù)　出處：《華東師范大學(xué)》2017年博士論文　論文類型：學(xué)位論文

【摘要】：本論文里,我們主要研究了一些特殊函數(shù)的相關(guān)問題。Gamma函數(shù)和Gamma函數(shù)比值的漸進展開問題,Theta函數(shù)高階導(dǎo)數(shù)在同余子群上的模形式結(jié)構(gòu)和相關(guān)應(yīng)用問題,得到了部分結(jié)果并給出了一些近似和恒等式。本文共分四個部分:第一部分:用Pade逼近的方法,利用Gamma函數(shù)的展開,在Laplace公式的基礎(chǔ)上建立了Gamma函數(shù)和Stirling公式更精確、更漂亮的近似和公式,在此基礎(chǔ)上我們給出了更一般的漸進結(jié)果,進一步確定了近似的參數(shù),我們把結(jié)果與近來著名的近似做比較,證明我們的結(jié)果是最好的。第二部分:利用Gamma函數(shù)的展開確定一個在特殊函數(shù)計算和階乘近似中有重要作用的Gamma函數(shù)比值Pn,建立了更精確的近似并確定了參數(shù),建立了新邊界,并給出了如何求得該比值更精確值的計算方法,進一步的證明了該比值類Stirling型和Laplace型的簡潔、精確的近似存在和結(jié)構(gòu)。第三部分:利用橢圓函數(shù)留數(shù)定理和Theta函數(shù)熱方程,由Theta函數(shù)的奇偶性、Theta函數(shù)的變換和Jacobi定理給出了 Theta函數(shù)高階導(dǎo)數(shù)的特值變換,使用冪級數(shù)展開的方法研究了Theta函數(shù)高階導(dǎo)數(shù)在同余子群上的模形式權(quán)變化。給出了Theta函數(shù)高階導(dǎo)數(shù)與一些特殊函數(shù)的恒等式。第四部分:利用橢圓函數(shù)留數(shù)定理和Theta函數(shù)熱方程,建立了Theta函數(shù)偶數(shù)階導(dǎo)數(shù)在同余子群上的一種結(jié)構(gòu),在此結(jié)構(gòu)中Theta函數(shù)導(dǎo)數(shù)在同余子群上模形式的權(quán)與階相等。進一步建立了Theta函數(shù)的偶數(shù)m階導(dǎo)數(shù)和Theta函數(shù)m次乘方的恒等結(jié)構(gòu),并給出Theta函數(shù)高階導(dǎo)數(shù)組合與Ochanine方程的系數(shù)參數(shù)化相關(guān)應(yīng)用。
[Abstract]:In this paper, we mainly study the problems related to some special functions. The asymptotic expansion of the ratio of Gamma function to Gamma function and the modular structure of the higher order derivative of the Theta function on the congruent subgroup and the related applications. Some results are obtained and some approximations and identities are given. This paper is divided into four parts: in the first part, by using the method of Pade approximation and the expansion of Gamma function, the Gamma function and Stirling formula are established on the basis of Laplace formula. On the basis of the more beautiful approximations and formulas, we give a more general asymptotic result, further determine the approximate parameters, and we compare the results with the recent well-known approximations. It is proved that our results are the best. Part two: using the expansion of Gamma function to determine a ratio of Gamma function that plays an important role in the calculation of special function and factorial approximation, establishing a more accurate approximation, determining parameters, and establishing a new boundary. The calculation method of the more accurate value of the ratio is given, and the succinct, exact approximate existence and structure of the Stirling type and Laplace type of the ratio are further proved. Part 3: using the elliptic function residue theorem and the Theta function heat equation, The special value transformation of higher derivative of Theta function is given by the transformation of Theta function and Jacobi theorem of Theta function. Using the method of power series expansion, we study the variation of the modulus weight of higher order derivatives of Theta functions on congruent subgroups. The identities of higher order derivatives of Theta functions and some special functions are given. Part 4th: using elliptic function residue determinations. Theorem and Theta function heat equation, In this paper, a structure of even order derivatives of Theta functions on congruent subgroups is established. In this structure, the weights and orders of the norm form of the derivatives of Theta functions are equal to those of the subgroups of congruence. Furthermore, the identity structures of the even m-order derivatives of Theta functions and the m multiplicators of Theta functions are established. The application of the combination of higher order derivatives of Theta function and the coefficient parameterization of Ochanine equation is also given.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O174.6

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本文編號：1613035