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] images

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

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

where γ is the Euler constant,

images

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

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