On power series of products of special functions

✍ Scribed by Yu.A. Brychkov

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 204 KB
• Volume: 24
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2011.03.013

✦ Synopsis

Expansion of functions in series with respect to special systems of functions is one of the principal tools of the theoretical and applied mathematics. The set of power functions z n , where n = 0, 1, 2 . . ., is used most often. The corresponding expansions for elementary and special functions are classical. Usually these are expansions of a single function. A few expansions of products of two functions can be found in the literature (see [1-4]), for example those of J µ (z)J ν (z), arcsin 2 z, arctan 2 z, ln(z 2 + 1) arctan z, e z 2 erf(z) and of some other functions. There are also several expansions of powers of elementary functions, for example sin ν z, cos ν z, ln n (z + 1) [2]. In this note, we give some methods of derivation of power I ν (z) is the modified Bessel function, C ν n (z) is the Gegenbauer polynomial.