Contents

• 1 Introduction
  • 1.1 Notations
  • 1.2 About the Gamma function
  • 1.3 Revisiting the hypergeom function
• 2 Orthogonal polynomials over $\left[ -1,\, +1\right] \protect$
  • 2.1 Starting from a self-adjoint operator
  • 2.2 The general three terms formula
  • 2.3 Closed formula for the $F_{n}$
  • 2.4 Normalization
  • 2.5 Generating functions
  • 2.6 About derivatives
  • 2.7 About the Chebyschev's generating function
  • 2.8 About the Legendre's generating function
• 3 The Markov-Bernstein majorant
  • 3.1 What is to solve
  • 3.2 Recurence over the $\chi _{n}$
  • 3.3 Factorisation of the recurrence relation
  • 3.4 Even and odd numbers
  • 3.5 Some results
• Bibliography
• Contents