Let be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞ , . Then, for any non-negative integer n, . When , this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of allow one to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm119-2-6, author = {Giacomo Gigante}, title = {A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type}, journal = {Colloquium Mathematicae}, volume = {120}, year = {2010}, pages = {237-254}, zbl = {1197.33010}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm119-2-6} }
Giacomo Gigante. A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type. Colloquium Mathematicae, Tome 120 (2010) pp. 237-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm119-2-6/