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On the discrete analogues of Appell function $F_1$

Ravi Dwivedi, Vivek Sahai

Abstract

In the present paper, two discrete forms of Appell function $F_1$ are introduced and studied. We examine the first discrete form in detail and give the results directly for the second. We study their regions of convergence, differential properties, integral representations, recursion relations, finite and infinite sums. The results and identities obtained in the paper are believed to be new and may find applications in various branches of science.

On the discrete analogues of Appell function $F_1$

Abstract

In the present paper, two discrete forms of Appell function are introduced and studied. We examine the first discrete form in detail and give the results directly for the second. We study their regions of convergence, differential properties, integral representations, recursion relations, finite and infinite sums. The results and identities obtained in the paper are believed to be new and may find applications in various branches of science.
Paper Structure (12 sections, 4 theorems, 80 equations)

This paper contains 12 sections, 4 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and preliminaries
  3. Discrete Appell function $\mathcal{F}_1^{(1)}$
  4. Integral representations
  5. Differential and difference formulae
  6. Finite and infinite summation formulas
  7. Recursion Formulae
  8. The discrete Appell function $\mathcal{F}^{(2)}_1$
  9. Integral representations
  10. Differential formulae
  11. Finite and infinite summation formulae
  12. Conclusion

Key Result

Theorem 3.1

Let $a$, $b_1$, $b_2$, $c$, $t_1$ and $t_2$ be complex numbers. Then for $\vert x\vert < 1$, $\vert y\vert < 1$ and $k_1, k_2 \in \mathbb{N}$, the discrete Appell function defined in 3.1 can be represented in the following integral forms:

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof
  • Theorem 7.1