# Appendix 1 Auxiliary Results from Mathematical, Functional and Stochastic Analysis

## A1.1. Special functions

DEFINITION A1.1.– *The gamma function is defined by the integral*

[A1.1]

*which converges for any x* >0. Here are some basic properties of the gamma function:

- i) the gamma function Γ(
*x*) is positive, continuous and has continuous derivatives of all orders on (0, ∞); - ii) ;
- iii);
- iv) the gamma function is logarithmically convex on (0, +∞), i.e. its logarithm is convex on (0, +∞);
- v) the Euler reflection formula is
[A1.2]
- vi) Legendre’s formula is
[A1.3]
- vii) Gauss’s formula for logarithmic derivative of gamma function has the following form:

where γ is the Euler constant,

DEFINITION A1.2.– *The beta function, also called the Euler integral of the first kind, is defined by the ...*

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