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21 pages, 328 KB

Open AccessArticle

Reduction for a Terminating Bivariate Hypergeometric Appell Series ℱ₁ (II)

by Mohamed Jalel Attia

Mathematics 2026, 14(11), 2021; https://doi.org/10.3390/math14112021 - 5 Jun 2026

Viewed by 176

This paper studies a terminating (“modified”) Appell function

F_{1}^{*} (α, β, β^{'}, β + β^{'}; x, y),

with

α \in Z_{\geq 1}

and [...] Read more.

This paper studies a terminating (“modified”) Appell function

F_{1}^{*} (α, β, β^{'}, β + β^{'}; x, y),

with

α \in Z_{\geq 1}

and

β, β^{'} \in Z_{\leq - 1}

, together with the associated terminating Gauss function

{}_{2}F_{1}^{*} (\begin{matrix} α, β \\ β + β^{'} \end{matrix}; z) .

The reduction formula for the Appell function

F_{1}^{*} (α, β, β^{'}, β + β^{'}; x, y)

\frac{1}{{(1 - y)}^{α}} {}_{2}F_{1} (\begin{matrix} α, β \\ β + β^{'} \end{matrix}; \frac{x - y}{1 - y})

breaks down when

(β, β^{'})

are integers less than or equal to

- 1

and it needs to be substituted with a revised identity that includes an explicit additional correction term

{CORR}^{(α, β, β^{'})} (x, y)

. This correction term is initially computed for specific cases (particularly for

α = 1, 2, 3

) and subsequently formulated and verified generally through mathematical induction on

α

. The final expression demonstrates a structured pattern reminiscent of binomial/Pascal coefficients and leads to various corollaries, including simplified boundary scenarios (such as when

α = 1

). Full article

(This article belongs to the Special Issue Combinatorics, Riordan Matrices and Umbral Calculus—in Memory of Prof. Emanuele Munarini)

18 pages, 960 KB

Open AccessArticle

Hybrid Algorithm via Reciprocal-Argument Transformation for Efficient Gauss Hypergeometric Evaluation in Wireless Networks

by Jianping Cai and Zuobin Ying

Mathematics 2025, 13(15), 2354; https://doi.org/10.3390/math13152354 - 23 Jul 2025

Viewed by 670

Abstract

The rapid densification of wireless networks demands efficient evaluation of special functions underpinning system-level performance metrics. To facilitate research, we introduce a computational framework tailored for the zero-balanced Gauss hypergeometric function [...] Read more.

Ψ (x, y) ≜ {}_{2}F_{1} (1, x; 1 + x; - y)

, a fundamental mathematical kernel emerging in Signal-to-Interference-plus-Noise Ratio (SINR) coverage analysis of non-uniform cellular deployments. Specifically, we propose a novel Reciprocal-Argument Transformation Algorithm (RTA), derived rigorously from a Mellin–Barnes reciprocal-argument identity, achieving geometric convergence with

O (1 / y)

. By integrating RTA with a Pfaff-series solver into a hybrid algorithm guided by a golden-ratio switching criterion, our approach ensures optimal efficiency and numerical stability. Comprehensive validation demonstrates that the hybrid algorithm reliably attains machine-precision accuracy (

\sim 10^{- 16}

) within 1 μs per evaluation, dramatically accelerating calculations in realistic scenarios from hours to fractions of a second. Consequently, our method significantly enhances the feasibility of tractable optimization in ultra-dense non-uniform cellular networks, bridging the computational gap in large-scale wireless performance modeling. Full article

(This article belongs to the Special Issue Advances in High-Performance Computing, Optimization and Simulation)

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17 pages, 320 KB

Open AccessArticle

On Summations of Generalized Hypergeometric Functions with Integral Parameter Differences

by Kirill Bakhtin and Elena Prilepkina

Mathematics 2024, 12(11), 1656; https://doi.org/10.3390/math12111656 - 25 May 2024

Viewed by 1653

Abstract

In this paper, we present an extension of the Karlsson–Minton summation formula for a generalized hypergeometric function with integral parameter differences. Namely, we extend one single negative difference in Karlsson–Minton formula to a finite number of integral negative differences, some of which will be repeated. Next, we continue our study of the generalized hypergeometric function evaluated at unity and with integral positive differences (IPD hypergeometric function at the unit argument). We obtain a recurrence relation that reduces the IPD hypergeometric function at the unit argument to

{}_{4}F_{3}

. Finally, we note that Euler–Pfaff-type transformations are always based on summation formulas for finite hypergeometric functions, and we give a number of examples. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)

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