Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bicomplex generalized hypergeometric functions and their applications

Snehasis Bera, Sourav Das, Abhijit Banerjee

TL;DR

This work extends generalized hypergeometric functions to bicomplex arguments by defining ${}_{p}F_{q}$ on $Z\in\mathbb{BC}$ and establishing its idempotent decomposition, convergence criteria, and integral representations. It derives differential relations, contiguous and quadratic transformation identities, and a unifying differential equation that generalizes classical results to the bicomplex setting. A key contribution is the construction of bicomplex generalized hypergeometric coherent states (BGHS) in a bicomplex Hilbert space, including ladder-operator structure and normalization via ${}_pF_q$ so that BGHS are eigenstates of an annihilation operator with bicomplex eigenvalue $Z$. The results subsume standard BC functions as special cases and pave the way for BC-valued techniques in quantum information and quantum optics contexts. Overall, the paper builds a robust BC-analytic framework for hypergeometric-type functions and demonstrates a clear pathway to applications in coherent-state quantum systems.

Abstract

In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation, differential relation, analyticity and contiguous relations of this function are derived. Additionally, applications in quantum information system and quantum optics are provided as a consequence.

Bicomplex generalized hypergeometric functions and their applications

TL;DR

This work extends generalized hypergeometric functions to bicomplex arguments by defining on and establishing its idempotent decomposition, convergence criteria, and integral representations. It derives differential relations, contiguous and quadratic transformation identities, and a unifying differential equation that generalizes classical results to the bicomplex setting. A key contribution is the construction of bicomplex generalized hypergeometric coherent states (BGHS) in a bicomplex Hilbert space, including ladder-operator structure and normalization via so that BGHS are eigenstates of an annihilation operator with bicomplex eigenvalue . The results subsume standard BC functions as special cases and pave the way for BC-valued techniques in quantum information and quantum optics contexts. Overall, the paper builds a robust BC-analytic framework for hypergeometric-type functions and demonstrates a clear pathway to applications in coherent-state quantum systems.

Abstract

In this work, generalized hypergeometric functions for bicomplex argument is introduced and its convergence criteria is derived. Furthermore, integral representation of this function has been established. Moreover, quadratic transformation, differential relation, analyticity and contiguous relations of this function are derived. Additionally, applications in quantum information system and quantum optics are provided as a consequence.
Paper Structure (9 sections, 27 theorems, 143 equations, 1 figure)

This paper contains 9 sections, 27 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bicomplex generalized hypergeometric function
  3. Integral representations of the bicomplex generalized hypergeometric function
  4. Some quadratic transformations
  5. Differential relation and analyticity of the bicomplex generalized hypergeometric function
  6. Contiguous relations
  7. Bicomplex generalized hypergeometric differential equation
  8. Applications of bicomplex generalized hypergeometric functions in coherent states
  9. Conclusion

Key Result

Theorem 1.1

bc_number A function $F:\Omega\subset \mathbb{BC} \rightarrow \mathbb{BC}$, where $F(Z)=f_1(z,z')+i_2f_2(z,z')$ is $\mathbb{BC}-holomorphic$ if and only if, $f_1$ and $f_2$ are holomorphic in $\Omega$ and satisfies following bicomplex Cauchy-Riemann equations :

Figures (1)

  • Figure 1:

Theorems & Definitions (53)

  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Saalschutz' theorem
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 43 more