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Jing-Feng Tian

Publications and source records attributed to Jing-Feng Tian.

11 recordsLinked to original sources
arXiv2026

Recurrence relations for the Maclaurin coefficients of products of elementary functions and Hypergeometric functions

In this paper, we investigate the recurrence relations for the Maclaurin coefficients of the products of elementary functions and hypergeometric functions. Specifically, we focus on the confluent hypergeometric function $\mathcal{M}(z) = h(z) M(a,c;z)$ and the Gaussian hypergeometric function $\mathcal{F}(z) = h(z) F(a,b;c;z)$, considering several specific choices for the function $h(z)$. In particular, we explore cases where $h(z)$ is chosen as $e^{pz}$, $(1-θz)^p$, $e^{-p \arctan z}$, $\sin(pz)$, $\cos(pz)$, $\sinh(pz)$, $\cosh(pz)$, $\arcsin(pz)$, and $\arccos(pz)$.

math.CV↗
arXiv2026

Recurrence Relations for the Maclaurin Coefficients of Products of Elementary Functions and the Bessel Functions

In this paper, we investigate recurrence relations for the Maclaurin coefficients of the products of a elementary function and the Bessel function of the first kind $\mathcal{J}(z) = h(z) J_ν(z)$ and the modified Bessel function of the first kind $\mathcal{I}(z) = h(z) I_ν(z)$ in the complex plane corresponding to several specific choices of $h(z)$. In particular, we specialize $h(z)$ as $e^{pz}$, $(1-θz)^p$, $e^{-p \arctan z}$, $\sin(pz)$, $\cos(pz)$, $\sinh(pz)$, $\cosh(pz)$, $\arcsin(pz)$ and $\arccos(pz)$.

math.CV↗
arXiv2026

Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane

For $a,b,c,z,p, θ\in \mathbb{C}$, where $\mathbb{C}$ is the complex plane, $-c\notin \mathbb{N\cup }\left\{ 0\right\} $, let \begin{equation*} \mathcal{M}\left( z\right) =\left( 1-θz\right) ^{p}M\left(a;c;z\right) =\sum_{n=0}^{\infty }u_{n}z^{n}, \end{equation*} where $|z| <\frac{1}θ$, $|\arg (1-θz)| < π$, and let \begin{equation*} \mathcal{G}\left( z\right) =(1-θz) ^{p}F(a,b;c;z) =\sum_{n=0}^{\infty }v_{n} z^{n}, \end{equation*} where $|z| < 1$, $|\arg (1-θz)| < π$. In this paper, we prove that the coefficients $u_{n}$ and $v_{n}$ for $n\geq 0$ satisfy a 3-order recurrence relation. These offer a new way to study confluent hypergeometric function $M(a;c;z)$ and Gauss hypergeometric function $F(a,b;c;z)$. And we provide other special functions' recurrence relations of their coefficients, such as error function, Bessel function, incomplete gamma function, complete elliptic integral and Chebyshev polynomials.

math.CV↗
arXiv2026

Recurrence relations and applications for the Maclaurin coefficients of squared and cubic hypergeometric functions

In this paper, we present and prove that the coefficients $u_n$ and $v_n$ in the series expansions $F^2(a,b;c;z) = \sum_{n=0}^\infty u_n z^n$ and $F^3(a,b;c;z) = \sum_{n=0}^\infty v_n z^n$ ($a,b,c,z \in \mathbb{C}$ and $-c \notin \mathbb{N} \cup \{0\}$) satisfy second- and third-order linear recurrence relations, respectively, where $F(a,b;c;x)$ denotes the Gaussian hypergeometric function and $\mathbb{C}$ is the complex plane. Our results provide recurrence relations for the Maclaurin coefficients of the squares and cubes of several classical special functions in the complex domain, including zero-balanced Gauss hypergeometric functions, elliptic integrals, as well as classical orthogonal polynomials such as Chebyshev, Legendre, Gegenbauer, and Jacobi polynomials. As applications, we first establish the monotonicity of a function involving Gauss hypergeometric functions and then present a new proof of the well-known Clausen's formula.

math.CA↗
arXiv2024

Monotonicity rules for the ratio of power series

In this paper, we present some monotonicity rules for the ratio of two power series $x\mapsto \sum_{k=0}^\infty a_k x^k / \sum_{k=0}^\infty b_k x^k$ under the assumption that the monotonicity of the sequence ${a_k/b_k}$ changes twice. Additionally, we introduce a local monotonicity rule in this paper.

math.CA↗
arXiv2023

Some monotonicity rules for quotient of integrals on time scales

As an efficient mathematical tool, monotonicity rules play an extremely crucial role in the real analysis field. In this paper, we explore some monotonicity rules for quotient of Delta, Nabla and Diamond-Alpha integrals with variable upper limits and parameters on time scales, respectively. Moreover, we consider the monotonicity rules for quotient of the product of multiple Delta integrals with parameters on time scales. Power series is also concerned for being a special case of integral with parameters on time scales.

math.CA↗
arXiv2023

Monotonicity rules for the ratio of two function series and two integral transforms

In this paper, we investigate the monotonicity of the functions $t \mapsto \frac{\sum_{k=0}^\infty a_k w_k(t)}{\sum_{k=0}^\infty b_k w_k(t)}$ and $x \mapsto \frac{\int_α^βf(t) w(t,x) \textrm{d} t}{\int_α^βg(t) w(t,x) \textrm{d} t}$, focusing on case where the monotonicity of $a_k/b_k$ and $f(t)/g(t)$ change once. The results presented also provide insights into the monotonicity of the ratios of two power series, two $\mathcal{Z}$-transforms, two discrete Laplace transforms, two discrete Mellin transforms, two Laplace transforms, and two Mellin transforms. Finally, we employ these monotonicity rules to present several applications in the realm of special functions and stochastic orders.

math.GM↗
arXiv2018

The monotonicity rules for the ratio of two Laplace transforms with applications

Let $f$ and $g$ be both continuous functions on $\left( 0,\infty \right) $ with $g\left( t\right) >0$ for $t\in \left( 0,\infty \right) $ and let $ F\left( x\right) =\mathcal{L}\left( f\right) $, $G\left( x\right) =\mathcal{L }\left( g\right) $ be respectively the Laplace transforms of $f$ and $g$ converging for $x>0$. We prove that if there is a $t^{\ast }\in \left( 0,\infty \right) $ such that $f/g$ is strictly increasing on $\left( 0,t^{\ast }\right) $ and strictly decreasing on $\left( t^{\ast },\infty \right) $, then the ratio $F/G$ is decreasing on $\left( 0,\infty \right) $ if and only if \begin{equation*} H_{F,G}\left( 0^{+}\right) =\lim_{x\rightarrow 0^{+}}\left( \frac{F^{\prime }\left( x\right) }{G^{\prime }\left( x\right) }G\left( x\right) -F\left( x\right) \right) \geq 0, \end{equation*} with \begin{equation*} \lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right) } =\lim_{t\rightarrow \infty }\frac{f\left( t\right) }{g\left( t\right) }\text{ \ and \ }\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left( x\right) }=\lim_{t\rightarrow 0^{+}}\frac{f\left( t\right) }{g\left( t\right) } \end{equation*} provide the indicated limits exist. While $H_{F,G}\left( 0^{+}\right) <0$, there is at leas one $x^{\ast }>0$ such that $F/G$ is increasing on $\left( 0,x^{\ast }\right) $ and decreasing on $\left( x^{\ast },\infty \right) $. As applications of this monotonicity rule, a unified treatment for certain bounds of psi function is presented, and some properties of the modified Bessel functions of the second are established. These show that the monotonicity rules in this paper may contribute to study for certain special functions because many special functions can be expressed as corresponding Laplace transforms.

math.CA↗
arXiv2017

Two asymptotic expansions for gamma function developed by Windschitl's formula

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} Γ\left( x+1\right) =\sqrt{2πx}\left( \tfrac{x}{e}\right) ^{x}\left( x\sinh \tfrac{1}{x}\right) ^{x/2}\exp \left( \sum_{k=3}^{n-1}\tfrac{\left( 2k\left( 2k-2\right) !-2^{2k-1}\right) B_{2k}}{2k\left( 2k\right) !x^{2k-1}} +R_{n}\left( x\right) \right) \end{equation*} with \begin{equation*} \left| R_{n}\left( x\right) \right| \leq \frac{\left| B_{2n}\right| }{2n\left( 2n-1\right) }\frac{1}{x^{2n-1}} \end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.

math.CA↗
arXiv2017

An accurate approximation formula for gamma function

In this paper, we present a very accurate approximation for gamma function: \begin{equation*} Γ\left( x+1\right) \thicksim \sqrt{2πx}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left( \frac{7}{324}\frac{1}{ x^{3}\left( 35x^{2}+33\right) }\right) =W_{2}\left( x\right) \end{equation*} as $x\rightarrow \infty $, and prove that the function $x\mapsto \ln Γ\left( x+1\right) -\ln W_{2}\left( x\right) $ is strictly decreasing and convex from $\left( 1,\infty \right) $ onto $\left( 0,β\right) $, where \begin{equation*} β=\frac{22\,025}{22\,032}-\ln \sqrt{2π\sinh 1}\approx 0.00002407. \end{equation*}

math.CA↗