Orthogonality relations for the associated Legendre functions of imaginary order

Abstract

Orthogonality relations for the associated Legendre functions of imaginary order are derived. They are expressed in terms of the Dirac delta function. The method is based on some known properties of the associated Legendre functions and the Dirac delta distribution. A special case of one of the relations has appeared in some recent applications.

Keywords:

• associated Legendre functions
• orthogonal functions
• Dirac delta distribution

MSC 2010 :

• 33C05
• 33C45

Acknowledgements

I wish to thank Prof. R. Szmytkowski for many useful discussions.

Notes

In fact, in the orthogonality formula (A.16) given by Hutasoit et al. 5 the term 2 π is missing. There is probably a missprint because the form of the equation (A.17) indicates that the factor 2 π must be included in (A.16).