arXiv subjects
Publications and source records attributed to Min-Jie Luo.
In this note, we revisit one of Erdélyi's product formulas for the univariate confluent hypergeometric function ${}_{1}F_{1}$ and extend it to the multivariate confluent hypergeometric functions $Φ_2^{(k)}$, $Ψ_2^{(k)}$, as well as to a certain matrix-variate confluent hypergeometric function. A useful connection related to fractional calculus is also given.
In this paper, we study various properties of Erdélyi's multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(α)}(x_1,\cdots,x_k)$ including their generating functions, product formulas and fractional integral representations. Many useful consequences are derived. New insights into various classical results including the relationships of the multivariate Laguerre polynomials with Oshima's fractional calculus operator and with an integral formula of Srivastava and Niukkanen are also mentioned. We further present an interesting evaluation for a generating function of the main diagonal sequence $L_{n,\cdots,n}^{(-β-kn)}(x_1,\cdots,x_k)$ which involves in a natural way the well-known Le Roy function ([Darboux Bull. 24 (2) (1899), 245--268]; [Toulouse Ann. 2 (2) (1900), 317--430]). The significance of the multivariate Laguerre polynomials $L_{n_1,\cdots,n_k}^{(α)}(x_1,\cdots,x_k)$ is demonstrated by observing that this class not only includes the generalized Hardy-Hille formula and the product formula but also contains the multiple Laguerre polynomials of the second kind as its important special cases. We briefly indicate also possible lines of future work.
This paper systematically studies the asymptotics of Humbert's bivariate confluent hypergeometric function $Φ_1[a,b;c;x, y]$. Specifically, we establish explicit asymptotic expansions in five distinct regimes: (i) $x\to\infty$; (ii) $y\to\infty$; (iii) $x\to\infty,\,y\to\infty$; (iv) $x$ or $y$ small, $xy$ fixed; and (v) $x\to 1$, $y$ fixed. The utility of these expansions is illustrated through concrete applications in the theory of Saran's hypergeometric function $F_M$, the Glauber-Ising model, and the theory of Prabhakar-type fractional integral operators. Several potential directions for future work are also outlined.
In this paper, we revisit the recent result of Luo, Xu, and Raina [Fractal Fract. 6 (3) (2022)] on an Erdélyi-type integral for Saran's three-variable hypergeometric function $F_K$. We provide a new proof of this integral and derive an attractive new integral related to Appell's function $F_2$. A further extension on the $L$-variable $F_K$ function, which appears in physics, is also discussed. Furthermore, we prove various $q$-Erdélyi-type integrals for the $q$-analogue of the $F_K$-function. An interesting discrete analogue is also included. We also provide a valuable compilation of the sources for known Erdélyi-type integrals of many different hypergeometric functions in the Appendix.
In this paper, we derive certain formulas giving the Laplace transforms of two generalized fractional integral operators introduced recently in [Fract. Calc. Appl. Anal. 20 (2) (2017), 422--446]. The main results provide generalizations to various known results. Some useful remarks related to the results presented in this paper are also mentioned.
In this paper, we establish the boundedness of the multiple Erdélyi-Kober fractional integral operators involving Fox's $H$-function on the Hardy space $H^1$. Our results generalize recent results of Kwok-Pun Ho [Proyecciones 39 (3) (2020), 663--677]. Some useful connections related to the Hausdorff operators are also mentioned.
We establish in this paper two inequalities for the multivariate Laguerre polynomials introduced and studied by Arthur Erdélyi [Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 146 (1937), 431--467]. These inequalities generalize the well-known Szegö's inequality for the Laguerre polynomials $L_n^{(α)}(x)$. We also mention briefly few insightful remarks giving a comparative analysis concerning the upper bounds of the derived inequalities in the concluding section.
In our recent work [SIGMA \textbf{20} (2024), 074, 13 pages], the leading behaviour of the Humbert function $Ψ_1[a,b;c,c';x,y]$ when $x\to\infty$ and $y\to +\infty$ has been derived in a direct and simple manner. In this paper, we obtain the complete asymptotics of $Ψ_1$ in the general case $x,y\to\infty$ along a new path. Indeed, our proof is based on a sharp estimate on ${}_2F_2[a,b-n;c,d-n;z]$, which is valid uniformly for $n\in\mathbb{Z}_{\geqslant 0}$ and large $z$.
A compilation of new results on the asymptotic behaviour of the Humbert functions $Ψ_1$ and $Ψ_2$, and also on the Appell function $F_2$, is presented. As a by-product, we confirm a conjectured limit which appeared recently in the study of the $1D$ Glauber-Ising model. We also propose two elementary asymptotic methods and confirm through some illustrative examples that both methods have great potential and can be applied to a large class of problems of asymptotic analysis. Finally, some directions of future research are pointed out in order to suggest ideas for further study.
For any $a\in\mathbb{C}$, the zeros of $ζ(s)-a$, denoted by $ρ_a=β_a+iγ_a$, are called $a$-points of the Riemann zeta function $ζ(s)$. In this paper, we reformulate some basic results about the $a$-points of $ζ(s)$ shown by Garunkštis and Steuding. We then deduce an asymptotic of the sum \[S_T(a,δ)=\sum_{τ<γ_a\leqslant T}ζ'(ρ_a+iδ)X^{ρ_a},\quad T\to\infty,\] where $0\neδ=\frac{2πα}{\log\frac{T}{2πX}}\ll 1$, and $X>0$ and $τ\geqslant|δ|+1$ are fixed. We also find the interesting varied behavior of $S_T(a,δ)$ in different $X$ ranges, which is more complicated than those described before by Gonek and Pearce-Crump.
Recently, Wald and Henkel (2018) derived the leading-order estimate of the Humbert functions $Φ_2$, $Φ_3$ and $Ξ_2$ for two large arguments, but their technique cannot handle the Humbert function $Ψ_1$. In this paper, we establish the leading asymptotic behavior of the Humbert function $Ψ_1$ for two large arguments. Our proof is based on a connection formula of the Gauss hypergeometric function and Nagel's approach (2004). This approach is also applied to deduce asymptotic expansions of the generalized hypergeometric function $_pF_q$ $(p\leqslant q)$ for large parameters, which are not contained in NIST handbook.
In this paper, we first establish asymptotic expansions of the Humbert function $Ψ_1$ for one large variable. The resulting expansions are then used to derive an asymptotic expansion of Saran's hypergeometric function $F_K$ when two of its variables become simultaneously large.