2.13 Hermite

Meixner-Pollaczek Hermite.

  
If we substitute in the definition (1.7.1) of the Meixner-Pollaczek polynomials and then let we obtain the Hermite polynomials :

Jacobi Hermite.

  
The Hermite polynomials given by (1.13.1) follow from the Jacobi polynomials defined by (1.8.1) by taking and letting in the following way :

Gegenbauer / Ultraspherical Hermite.

    
The Hermite polynomials given by (1.13.1) follow from the Gegenbauer (or ultraspherical) polynomials defined by (1.8.15) by taking and letting in the following way :

Krawtchouk Hermite.

  
The Hermite polynomials follow from the Krawtchouk polynomials defined by (1.10.1) by setting and then letting :

Laguerre Hermite.

  
The Hermite polynomials defined by (1.13.1) can be obtained from the Laguerre polynomials given by (1.11.1) by taking the limit in the following way :

Charlier Hermite.

  
If we set in the definition (1.12.1) of the Charlier polynomials and let we find the Hermite polynomials defined by (1.13.1). In fact we have