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Extending the Meijer $G$-function

Dmitrii Karp, Alexey Kuznetsov

Abstract

By replacing the Euler gamma function by the Barnes double gamma function in the definition of the Meijer $G$-function, we introduce a new family of special functions, which we call $K$-functions. This is a very general class of functions, which includes as special cases Meijer $G$-functions (thus also all hypergeometric functions ${}_p F_q$) as well as several new functions that appeared recently in the literature. Our goal is to define the $K$-function, study its analytic and transformation properties and relate it to several functions that appeared recently in the study of random processes and the fractional Laplacian. We further introduce a generalization of the Kilbas-Saigo function and show that it is a special case of $K$-function.

Extending the Meijer $G$-function

Abstract

By replacing the Euler gamma function by the Barnes double gamma function in the definition of the Meijer -function, we introduce a new family of special functions, which we call -functions. This is a very general class of functions, which includes as special cases Meijer -functions (thus also all hypergeometric functions ) as well as several new functions that appeared recently in the literature. Our goal is to define the -function, study its analytic and transformation properties and relate it to several functions that appeared recently in the study of random processes and the fractional Laplacian. We further introduce a generalization of the Kilbas-Saigo function and show that it is a special case of -function.
Paper Structure (13 sections, 16 theorems, 145 equations, 2 figures, 1 table)

This paper contains 13 sections, 16 theorems, 145 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Barnes double gamma function
  3. Definition of the Meijer-Barnes $K$ function
  4. Algebraic asymptotics of $K$ function
  5. The Mellin transform and integral equations satisfied by $K$ function
  6. Examples of $K$ function in applications
  7. Extrema of stable processes
  8. Eigenfunctions of fractional Laplacian on $(0,\infty)$
  9. Exponential functionals of hypergeometric processes
  10. Generalized Kilbas-Saigo function
  11. Barnes beta distributions
  12. Alternative definition in terms of $\Gamma_2(z;\omega_1,\omega_2)$
  13. Proof of \ref{['eq:G_s_over_tau_transformation']}

Key Result

Proposition 1

For $\textnormal{Re}(z)>0$ and $\textnormal{Re}(\tau)>0$ we have where In the above formula, we have $a_1(\tau)=-(1+\tau)/(2\tau)$, $a_2(\tau)=1/(2\tau)$, $b_2(\tau)=-(3/2+\ln(\tau) )/(2\tau)$ and the explicit expressions for other coefficients can be found in AK_2023Bill1997.

Figures (2)

  • Figure 1: Examples of contours ${\mathcal{L}}_{\theta^+,\theta^-}$, separating the sets $\Lambda^+$ (red diamonds) and $\Lambda^-$ (blue crosses) and having prescribed angles at infinity.
  • Figure 2: Mapping the complex plane by the function $\omega s$ (left graph) and $\omega s^2$ (right graph).

Theorems & Definitions (27)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Definition 1
  • Remark 1
  • Theorem 1: Analyticity properties
  • ...and 17 more