Properties of multiple Hermite and multiple Laguerre polynomials by the generating function

Abstract

A generating function is obtained for multiple Hermite and multiple Laguerre polynomials from the Rodrigues formula, and then interesting recurrence relations are given. We also find a relation between multiple orthogonal polynomials and ordinary orthogonal polynomials such as multiple Hermite polynomials with classical Hermite polynomials and multiple Laguerre polynomials with classical Laguerre polynomials. Lastly, we find a differential equation for the multiple orthogonal polynomials by the generating function, which is a reconstruction of the works by Aptekarev et al. [Aptekarev, A.I., Branquinho, A., Van Assche, W., 2003, Multiple orthogonal polynomials for classical weights. Transaction of American Mathematical Society, 355, 3887–3914.].

Keywords:

• Multiple orthogonal polynomials
• Generating function
• Rodrigues’ formula
• Differential equation

Acknowledgements

The author wishes to thank Walter Van Assche for the problem suggestions and valuable comments on this topic while visiting Katholieke Universteit Leuven in Belguim. This work was supported by Korea Research Foundation Grant (KRF-2003-015-C00022).

Additional information

Notes on contributors

D. W. Lee

Email: [email protected]