Eulerian functions. Classical orthogonal polynomials

Eulerian functions are in a special position: they appear in almost all questions concerning other special functions, i.e. they are involved, in particular the Gamma function, in most problems arising from mathematical physics. It therefore seems necessary to study these functions before any others.

Historically, the gamma function arose from the need to give meaning to x! for any complex x. Stirling's formula, providing an estimate of x! for large x, fundamental in questions of asymptotic behavior, completes the primordial status of the Γ function.

From the outset, its study involves the fundamental principles of the theory of functions of complex variables. Remarkably, the justification of its main properties can be set out in elementary terms, without ceasing to be rigorous.

Long numbering three, the classic orthogonal polynomial sequences, like the three musketeers, have in fact numbered four since 1949: Hermite polynomials, Laguerre polynomials (one-parameter), Bessel polynomials (one-parameter) and Jacobi polynomials (two-parameter). Bessel polynomials have been slow to achieve the status of classical polynomials because the Bessel form is not positive definite for any parameter value.

Algebraically, an orthogonal sequence is said to be classical if the sequence of derivatives is also orthogonal. With this definition, Bessel polynomials are classical. Other definitions are possible; the most important are described here.

Instead of a study based on the hypergeometric character of classical polynomials, we prefer a purely algebraic exposition, which has the merit of linking the various characterizations in a natural way. With this point of view, questions of shape representation are relegated to the background.