关于双基 Humbert 超几何函数 $Φ_1$

Certain new formulas for bibasic Humbert hypergeometric functions $Ψ_{1}$ and $Ψ_{2}$

The main aim of the present work is to give some interesting the $q$-analogues of various $q$-recurrence relations, $q$-recursion formulas, $q$-partial derivative relations, $q$-integral representations, transformation and summation formulas for bibasic Humbert hypergeometric functions $Ψ_{1}$ and $Ψ_{2}$ on two independent bases $q$ and $p$ of two variables and some developments formulae, believed to be new, by using the conception of $q$-calculus. Finally, some interesting special cases and straightforward identities connected with bibasic Humbert hypergeometric series of the types $Ψ_{1}$ a…

基本Humbert超几何函数Φ1、Φ2与Φ3的推广

An extension of basic Humbert hypergeometric functions Φ1, Φ2 and Φ3

本文系统研究了基本Humbert合流超几何函数Φ1、Φ2和Φ3在参数约束下的q-偏导数公式、递推关系、偏微分方程、求和与变换公式及q-积分表示，推广了该领域的已知结果。

Humbert confluent hypergeometric函数关于其参数的导数

Derivatives of Humbert confluent hypergeometric functions with respect to their parameters

本文研究了七种经典Humbert函数关于其参数的导数，推导出显式公式并给出闭合微分和约简公式，同时展示了Taylor型参数展开的应用。

On a representation of Humbert's double hypergeometric series in a series of Gauss's 2F1 function

In this paper, we aim to obtain a representation of Humbert's hy- pergeometric function in a series of Gauss's function 2F1. A few interesting results have also been deduced as special case of our main findings.

两个大自变量下Humbert函数$Ψ_1$的渐近性

Asymptotics of the Humbert Function $Ψ_1$ for Two Large Arguments

本文利用Gauss超几何函数连接公式与Nagel方法，首次建立了Humbert函数$Ψ_1$在两个大自变量下的首阶渐近行为，并将该方法推广到NIST手册未涵盖的广义超几何函数$_pF_q$（$p\leqslant q$）的大参数渐近展开。

The asymptotic expansion of the Humbert hyper-Bessel function

We consider the asymptotic expansion of the Humbert hyper-Bessel function expressed in terms of a ${}_0F_2$ hypergeometric function by \[J_{m,n}(x)=\frac{(x/3)^{m+n}}{m! n!}\,{}_0F_2(-\!\!\!-;m+1, n+1; -(x/3)^3)\] as $x\to+\infty$, where $m$, $n$ are not necessarily non-negative integers. Particular attention is paid to the determination of the exponentially small contribution. The main approach utilised is that described by the author (J. Comput. Appl. Math. {\bf 234} (2010) 488-504); a leading-order estimate is also obtained by application of the saddle-point method applied to an integral re…