一些特殊函数的若干问题研究

发布时间：2018-03-14 21:29

本文选题：Gamma函数　切入点：ψ函数　出处：《华东师范大学》2017年博士论文　论文类型：学位论文

【摘要】：本论文里,我们主要研究了一些特殊函数的相关问题。Gamma函数和Gamma函数比值的渐进展开问题,Theta函数高阶导数在同余子群上的模形式结构和相关应用问题,得到了部分结果并给出了一些近似和恒等式。本文共分四个部分:第一部分:用Pade逼近的方法,利用Gamma函数的展开,在Laplace公式的基础上建立了Gamma函数和Stirling公式更精确、更漂亮的近似和公式,在此基础上我们给出了更一般的渐进结果,进一步确定了近似的参数,我们把结果与近来著名的近似做比较,证明我们的结果是最好的。第二部分:利用Gamma函数的展开确定一个在特殊函数计算和阶乘近似中有重要作用的Gamma函数比值Pn,建立了更精确的近似并确定了参数,建立了新边界,并给出了如何求得该比值更精确值的计算方法,进一步的证明了该比值类Stirling型和Laplace型的简洁、精确的近似存在和结构。第三部分:利用椭圆函数留数定理和Theta函数热方程,由Theta函数的奇偶性、Theta函数的变换和Jacobi定理给出了 Theta函数高阶导数的特值变换,使用幂级数展开的方法研究了Theta函数高阶导数在同余子群上的模形式权变化。给出了Theta函数高阶导数与一些特殊函数的恒等式。第四部分:利用椭圆函数留数定理和Theta函数热方程,建立了Theta函数偶数阶导数在同余子群上的一种结构,在此结构中Theta函数导数在同余子群上模形式的权与阶相等。进一步建立了Theta函数的偶数m阶导数和Theta函数m次乘方的恒等结构,并给出Theta函数高阶导数组合与Ochanine方程的系数参数化相关应用。
[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.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O174.6

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