# 数学代写|数值方法作业代写numerical methods代考|MATH861

## 数学代写|数值方法作业代写numerical methods代考|k-Hypergeometric Series Solutions

It is well known that many phenomena in physical and technical applications are governed by a variety of differential equations. We should notice that these differential equations have appeared in many different research fields, for instance in the theory of automorphic function, in conformal mapping theory, in the theory of representations of Lie algebras, and in the theory of difference equations. Analytical and numerical methods to solve ordinary differential equations are ancient and interesting research direction in differentiable dynamical systems and their applications. Let us consider a so-called non-homogeneous $k$-hypergeometric differential equation of the form:
$$k z(1-k z) \frac{d^2 y}{d z^2}+[c-(k+a+b) k z] \frac{d y}{d z}-a b y=f(z)$$
with the independent variable $z$, where $a, b, c, k$ are several constants with $a, b, c \in \mathbb{R}, k \in \mathbb{R}^{+}$, and the function $f(z)$ is holomorphic in an interval $\mathcal{D} \subseteq \mathbb{C}$. In the case of $k=1$, if the function $f(z)$ vanishes identically, then Equation (1) degrades into a linear homogeneous hypergeometric ordinary differential equation presented by Euler [1] in 1769, which has the following normalized form:
$$z(1-z) \frac{d^2 y}{d z^2}+[c-(1+a+b) z] \frac{d y}{d z}-a b y=0 ;$$
such an equation has been extensively studied.
The solutions of a differential equation relate to many absorbing special functions in mathematics, physics, and engineering. For instance, the solution could be presented by power series [2,3], continued fraction [4-6], zeta function [7-10], and hypergeometric series [11-16]. Among these special functions, the hypergeometric series, denoted by:
$${ }2 F_1[a, b ; c ; z]=\sum{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n n !} z^n,$$ can be applied to the solution of the differential Equation (2). For Equation (2), a hypergeometric series solution ${ }_2 F_1$ can be derived by the Frobenius method. The so-called hypergeometric series was researched firstly by Wallis [11] in 1655. Since then, Euler, too, had researched the topic on the hypergeometric series, but the first full systematic study was introduced by Gauss [12]. Some works and complete references concerning both the hypergeometric series and the certain equation (2) can be found in Kummer [13], Riemann [14], Bailey [15,16], Chaundy [17], Srivastava [18], Whittaker [19], Beukers [20], Gasper [21], Olde Daalhuis [22,23], Dwork [24], Chu [25], Yilmazer et al. [26], Morita et al. [27], Abramov et al. [28], Alfedeel et al. [29], and the literature therein. However, in contrast to the extensive studies on Equation (2), other hypergeometric differential equations with $k \in \mathbb{R}^{+}$are very limited.

## 数学代写|数值方法作业代写numerical methods代考|Preliminaries

In this section, we briefly review some basic definitions and facts concerning the $k$-hypergeometric series and the ordinary differential equation. Some surveys and literature for $k$-hypergeometric series and the $k$-hypergeometric differential equation can be found in Díaz et al. [30,31], Krasniqi [32,33], and Mubeen et al. [38,39].

Definition 1. Assume that $x \in \mathbb{C}, k \in \mathbb{R}^{+}$and $n \in \mathbb{N}^{+}$, then the Pochhammer $k$-symbol $(x){n, k}$ is defined by: $$(x){n, k}=x(x+k)(x+2 k) \ldots[x+(n-1) k] .$$
In particular, we denote $(x){0, k} \equiv 1$. Therefore, we have the following facts: (i) $(x){n+1, k}=(x+n k)(x){n, k}$. (ii) $(1){n, 1}=n ! ; \quad\left(\frac{1}{2}\right){n, 1}=\frac{(2 n-1) ! !}{2^n} ; \quad\left(\frac{3}{2}\right){n, 1}=\frac{(2 n+1) ! !}{2^n}$.
(iii) $(x){n, 1}=\frac{\Gamma(x+n)}{\Gamma(x)}$, where $\Gamma(x)$ is the Gamma function defined by $\int_0^{\infty} e^{-t} t^{x-1} d t$. (iv) $(1){n, 2}=(2 n-1) ! ! ;(2){n, 2}=(2 n) ! ! ;(3){n, 2}=(2 n+1) ! ! ;(4){n, 2}=\frac{(2 n+2) ! !}{2}$. Definition 2. Assume that $a, b, c \in \mathbb{C}, k \in \mathbb{R}^{+}$and $n \in \mathbb{N}^{+}$, then the $k$-hypergeometric series with three parameters $a, b$, and $c$ is defined as: $${ }_2 F{1, k}[(a, k),(b, k) ;(c, k) ; z]=\sum_{n=0}^{\infty} \frac{(a){n, k}(b){n, k}}{(c)_{n, k} k !} z^n,$$
where $c \neq 0,-1,-2,-3, \ldots$ and $z \in \mathbb{C}$.
Definition 3. Assume that $Y_0(z), Y_1(z)$, and $Y_2(z)$ are three functions of $z$. Let a second-order ordinary differential equation be written in the following form:
$$Y_2(z) \frac{d^2 y}{d z^2}\left|Y_1(z) \frac{d y}{d z}\right| Y_0(z)-0 .$$
Then, the method about finding an infinite series solution of Equation (7) is called the Frobenius method.

# 数值方法代考

## 数学代写|数值方法作业代写numerical methods代考|k-Hypergeometric Series Solutions

$$k z(1-k z) \frac{d^2 y}{d z^2}+[c-(k+a+b) k z] \frac{d y}{d z}-a b y=f(z)$$

$$z(1-z) \frac{d^2 y}{d z^2}+[c-(1+a+b) z] \frac{d y}{d z}-a b y=0$$

$$2 F_1[a, b ; c ; z]=\sum n=0^{\infty} \frac{(a)_n(b)_n}{(c)_n n !} z^n,$$

## 数学代写|数值方法作业代写numerical methods代考|Preliminaries

$$(x) n, k=x(x+k)(x+2 k) \ldots[x+(n-1) k] .$$

$\Leftrightarrow(x) n, 1=\frac{\Gamma(x+n)}{\Gamma(x)} ，$ 在哪里 $\Gamma(x)$ 是由下式定义的 Gamma 函数 $\int_0^{\infty} e^{-t} t^{x-1} d t$. (四)
(1) $n, 2=(2 n-1) ! ! ;(2) n, 2=(2 n) ! ! ;(3) n, 2=(2 n+1) ! ! ;(4) n, 2=\frac{(2 n+2) ! !}{2}$. 定义 2 . 假设 $a, b, c \in \mathbb{C}, k \in \mathbb{R}^{+}$和 $n \in \mathbb{N}^{+}$，那么 $k$ – 具有三个参数的超几何级数 $a, b$ ，和 $c$ 定义为:
$${ }2 F 1, k[(a, k),(b, k) ;(c, k) ; z]=\sum{n=0}^{\infty} \frac{(a) n, k(b) n, k}{(c)_{n, k} k !} z^n,$$

$$Y_2(z) \frac{d^2 y}{d z^2}\left|Y_1(z) \frac{d y}{d z}\right| Y_0(z)-0 .$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: