Abstract
Three types of special functions are considered: (1) Cylindrical functions, (2) Jacobi functions (this further subdivides into two subcases), and (3) Confluent hypergeometric functions. Some properties of these functions, which are needed in studying the expansions in terms of them, are discussed. In particular, various differentiation formulas, integral representations, and asymptotic estimates are presented. The key formulas are the Koornwinder integral representations for Jacobi functions and their analogues for the Kummer confluent hypergeometric function. These formulas generalize the classical Mehler–Dirichlet representation for Legendre functions and allow one to obtain Paley–Wiener-type theorems for the corresponding integral transforms.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag London Limited
About this chapter
Cite this chapter
Volchkov, V.V., Volchkov, V.V. (2009). Some Special Functions. In: Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84882-533-8_7
Download citation
DOI: https://doi.org/10.1007/978-1-84882-533-8_7
Publisher Name: Springer, London
Print ISBN: 978-1-84882-532-1
Online ISBN: 978-1-84882-533-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)