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April 8, 2026Open Access

Transcendental Operational Mathematics on Hypergeometric Function Families A Unified Iteration Theory from Single-Variable Hypergeometric Functions to Multivariable Appell Functions, q-Hypergeometric Functions, and Elliptic Hypergeometric Functions

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SLshifa liu

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Overview

Establishes a theoretical framework for hypergeometric functions, indicating a unified view of complex iterations.

Key Points

  • The aim is to develop a complete theoretical system of transcendental operational mathematics for hypergeometric function families.
  • Establishes axiomatic systems with twenty independent axioms.
  • Derives fractional-order series expansions and integral representations.
  • Constructs translation iterations for single-variable and multivariable functions.
  • Analyzes the singularity structure of complex-order iterations.
  • Proves categorical duality and equivalences in iterations.
  • Establishes a unified theory of fractional-order iterations.
  • Proves the transcendence of fractional-order hypergeometric values under certain conjectures.
  • Provides complete algorithms and numerical verification, transforming open problems into proven theorems.

Cite This Study

shifa liu (2025) studied this question.

synapsesocial.com/papers/69d5f0ee74eaea4b11a7a5cehttps://doi.org/10.5281/zenodo.19444215
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